AIA DEM Workshop NAOJMark Cheung LMSAL x NAOJ

cheunglmsalcom

Schedulebull Monday March 28th

bull 10 am - 12 noon Lecture (interactive)

bull 1pm onward Data analysis lab exercises

bull Tuesday March 29th Schedule is flexible

bull BUT 1pm to 230pm Shimojo-san ALMA Solar Town Hall

bull Data analysis lab exercises

bull Show and tell

bull ldquoOffice hoursrdquo

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

3

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

4

AimsBy the end of this workshop I hope you will have some answers to the following

bull How to find information about AIA data analysis

bull What do SDOAIA EUV images show

bull What is the differential emission measure (DEM) problem

bull How do we perform DEM inversions on SDOAIA data

bull How to perform quantitative investigations of the physical evolution of the solar corona

5

6

A taste of what we can do with AIA Below is a differential emission measure inversion

Useful links documentationSDO User Guidehttpwwwlmsalcomsdouserguidehtml (Does not yet include how to do DEMs) AIA DEM package (in beta)httptinyurlcomaiadem Documentation about AIAhttpswwwlmsalcomsdodocs bull AIA Instrument Paper (Lemen et al 2012) bull Initial Calibration of AIA (Boerner et al 2012) CHIANTI User Guide$SSWpackageschiantidoccugpdf

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

3

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

4

AimsBy the end of this workshop I hope you will have some answers to the following

bull How to find information about AIA data analysis

bull What do SDOAIA EUV images show

bull What is the differential emission measure (DEM) problem

bull How do we perform DEM inversions on SDOAIA data

bull How to perform quantitative investigations of the physical evolution of the solar corona

5

6

A taste of what we can do with AIA Below is a differential emission measure inversion

Useful links documentationSDO User Guidehttpwwwlmsalcomsdouserguidehtml (Does not yet include how to do DEMs) AIA DEM package (in beta)httptinyurlcomaiadem Documentation about AIAhttpswwwlmsalcomsdodocs bull AIA Instrument Paper (Lemen et al 2012) bull Initial Calibration of AIA (Boerner et al 2012) CHIANTI User Guide$SSWpackageschiantidoccugpdf

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

4

AimsBy the end of this workshop I hope you will have some answers to the following

bull How to find information about AIA data analysis

bull What do SDOAIA EUV images show

bull What is the differential emission measure (DEM) problem

bull How do we perform DEM inversions on SDOAIA data

bull How to perform quantitative investigations of the physical evolution of the solar corona

5

6

A taste of what we can do with AIA Below is a differential emission measure inversion

Useful links documentationSDO User Guidehttpwwwlmsalcomsdouserguidehtml (Does not yet include how to do DEMs) AIA DEM package (in beta)httptinyurlcomaiadem Documentation about AIAhttpswwwlmsalcomsdodocs bull AIA Instrument Paper (Lemen et al 2012) bull Initial Calibration of AIA (Boerner et al 2012) CHIANTI User Guide$SSWpackageschiantidoccugpdf

AimsBy the end of this workshop I hope you will have some answers to the following

bull How to find information about AIA data analysis

bull What do SDOAIA EUV images show

bull What is the differential emission measure (DEM) problem

bull How do we perform DEM inversions on SDOAIA data

bull How to perform quantitative investigations of the physical evolution of the solar corona

5

6

A taste of what we can do with AIA Below is a differential emission measure inversion

Useful links documentationSDO User Guidehttpwwwlmsalcomsdouserguidehtml (Does not yet include how to do DEMs) AIA DEM package (in beta)httptinyurlcomaiadem Documentation about AIAhttpswwwlmsalcomsdodocs bull AIA Instrument Paper (Lemen et al 2012) bull Initial Calibration of AIA (Boerner et al 2012) CHIANTI User Guide$SSWpackageschiantidoccugpdf

6

A taste of what we can do with AIA Below is a differential emission measure inversion

Useful links documentationSDO User Guidehttpwwwlmsalcomsdouserguidehtml (Does not yet include how to do DEMs) AIA DEM package (in beta)httptinyurlcomaiadem Documentation about AIAhttpswwwlmsalcomsdodocs bull AIA Instrument Paper (Lemen et al 2012) bull Initial Calibration of AIA (Boerner et al 2012) CHIANTI User Guide$SSWpackageschiantidoccugpdf

Useful links documentationSDO User Guidehttpwwwlmsalcomsdouserguidehtml (Does not yet include how to do DEMs) AIA DEM package (in beta)httptinyurlcomaiadem Documentation about AIAhttpswwwlmsalcomsdodocs bull AIA Instrument Paper (Lemen et al 2012) bull Initial Calibration of AIA (Boerner et al 2012) CHIANTI User Guide$SSWpackageschiantidoccugpdf

1 Introduction 11 Synopsis of the SDO Mission 12 About this Guide 13 Data Products from AIA 14 Data Products from EVE 15 Data Products from HMI 16 SDO Mission Data Policy 2 How to Browse SDO Data 21 ldquoThe Sun Nowrdquo 22 ldquoSun In Timerdquo 23 SolarMonitor 24 The Helioviewer Project 3 How to Find SDO Data 31 Heliophysics Events Knowledgebase 32 iSolSearch 4 How to Get AIA and HMI Data 41 How Data are Organized at the JSOC 42 The lookdata Tool 421 Getting Dataseries Names 422 Selecting Records 423 Exporting Data 424 JSOC Image Timing Details 43 The Cutout Service 44 The Virtual Solar Observatory 45 Synoptic Data 46 Uncompressing the Data 461 Compiling and Installing imcopy 5 How to Use SolarSoft with SDO Data 51 Installing or Upgrading SolarSoft 511 Doing a Clean Install 512 Updating Packages for an Existing Installation 513 Using a Proxy Server with SSW

52 Querying the HER 521 Basic Query Syntax 522 Adding Filters to Queries 53 Retrieving Data from the JSOC 531 Getting Dataseries Names 532 Selecting Records 533 Exporting Data 534 Exporting Specific Segments 54 Requesting a Cutout Using SSW 55 Using the SSW Interface to the VSO 551 Querying the VSO 552 Downloading Data from the VSO 56 Uncompressing the Image Data 561 Using read_sdopro 562 Using read_sdopro with a Shared Library 6 How to Process AIA Data 61 Using aia_preppro to Align and Derotate AIA Data 611 Aligning Full-Disk Images 612 Aligning Cropped Images and Cutouts 62 Co-Aligning HMI Data with AIA Data 63 Despiking andor Respiking AIA Data 64 Point Spread Function Deconvolution 65 Getting AIA Filter Response Data 66 Miscellaneous AIA Tasks 661 Using AIA Standard Scaling and Colors 662 Creating AIA Tri-Color Images 663 Plotting an AIA Light Curve 664 Interfacing with plot_mappro (Dom Zarrorsquos Mapping Software) 665 Checking the QUALITY Keyword 7 Frequently Asked Questions 8 Lists of Useful Links 9 Version History of this Guide 10 Contributors to this Guide

SDO User Guide

httpwwwlmsalcomsdouserguidehtml

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

9

What do we see in AIAEUV images

10

Name aia_get_response Purpose return SDOAIA instrument response data structures Input Parameters Output function returns AIA wavelength response spectral model or temperature response Descriptions of the structures returned by this routine are available via httpsohowwwnascomnasagovsolarsoftsdoaiaresponseREADMEtxt Calling Examples IDLgt effarea=aia_get_response(area dn) return per wavelength effective area IDLgt effarea=aia_get_response(areafulldn) same with component details (filterccd) IDLgt emiss=aia_get_response(emiss full) return default CHIANTI model with line list IDLgt tresp=aia_get_response(tempdnevenorm) return temperature response including constant scale factors to give good overall agreement with SDOEVE

What do EUV images from SDOAIA show

11

See section 65 of SDO User Guide for extensive explanation

What do we see in AIAEUV images

10

Name aia_get_response Purpose return SDOAIA instrument response data structures Input Parameters Output function returns AIA wavelength response spectral model or temperature response Descriptions of the structures returned by this routine are available via httpsohowwwnascomnasagovsolarsoftsdoaiaresponseREADMEtxt Calling Examples IDLgt effarea=aia_get_response(area dn) return per wavelength effective area IDLgt effarea=aia_get_response(areafulldn) same with component details (filterccd) IDLgt emiss=aia_get_response(emiss full) return default CHIANTI model with line list IDLgt tresp=aia_get_response(tempdnevenorm) return temperature response including constant scale factors to give good overall agreement with SDOEVE

What do EUV images from SDOAIA show

11

See section 65 of SDO User Guide for extensive explanation

Name aia_get_response Purpose return SDOAIA instrument response data structures Input Parameters Output function returns AIA wavelength response spectral model or temperature response Descriptions of the structures returned by this routine are available via httpsohowwwnascomnasagovsolarsoftsdoaiaresponseREADMEtxt Calling Examples IDLgt effarea=aia_get_response(area dn) return per wavelength effective area IDLgt effarea=aia_get_response(areafulldn) same with component details (filterccd) IDLgt emiss=aia_get_response(emiss full) return default CHIANTI model with line list IDLgt tresp=aia_get_response(tempdnevenorm) return temperature response including constant scale factors to give good overall agreement with SDOEVE

What do EUV images from SDOAIA show

11

See section 65 of SDO User Guide for extensive explanation

From aia_get_responsepro in SSW12

Brendan OrsquoDwyer et al SDOAIA response to coronal hole quiet Sun active region and flare plasma

Fig 1 Plot of DEM curves for coronal hole quiet Sun active regionand flare plasma (see text for references)

and AR spectra Density values of Ne = 2 times 108 cmminus3Ne = 5 times 108 cmminus3 and Ne = 5 times 109 cmminus3 were used in cal-culating the CH QS and AR spectra respectively These are thesame density values as those used to calculate the contributionfunctions for the lines which were used to constrain the DEMcurves for the CH QS and AR cases For the flare spectruma value of Ne = 1 times 1011 cmminus3 and the solar coronal abun-dances of Feldman et al (1992) were used The use of eitherphotospheric or coronal abundances in calculating the syntheticspectra reflects the original use of either photospheric or coronalabundances in generating the DEM curves These synthetic spec-tra were then convolved with the effective area of each channelThe effective areas were obtained from P Boerner (2009 privatecommunication) with the exception of the 171 Aring and 335 Aringchannels for which updated versions were used obtained fromSolarsoft (12 July 2010)

3 ResultsTable 1 lists those spectral lines which contribute more than 3to the total emission in each channel for CH QS AR and flareplasma Also included is the fractional contribution of the con-tinuum emission for any case where the continuum contributesmore than 3 to the total emission in a channel Synthetic spec-tra for each of the channels are displayed in Fig 2 - 8 For everychannel each spectrum has been divided by the peak intensityof the strongest spectrum Weaker spectra have been scaled byfactors indicated in each figure

The 94 Aring channel is expected1 to observe the Fe XVIII 9393Aringline [log T[K] sim 685] in flaring regions For both AR and flareplasma (see Fig 2) the dominant contribution comes from theFe XVIII 9393 Aring line However for the CH and QS spectrum (seeFig 2) the dominant contribution comes from the Fe X 9401 Aringline [log T sim 605]

In flaring regions the 131 Aring channel is expected1 to observethe Fe XX 13284 Aring and Fe XXIII 13291 Aring lines However forthe flare spectrum (see Fig 3) the dominant contribution comesfrom the Fe XXI 12875 Aring line [log T sim 705] The combined con-tribution of the Fe XX 13284 Aring and Fe XXIII 13291 Aring lines is lessthan ten percent of the total emission For CH observations the131 Aring channel is expected1 to be dominated by Fe VIII lines [logT sim 56] From our simulations the dominant contribution forCH and QS plasma (see Fig 3) does come from Fe VIII lines butwith a significant contribution from continuum emission For the

Table 1 Predicted AIA count rates

Ion λ T ap Fraction of total emissionAring K CH QS AR FL

94 Aring Mg VIII 9407 59 003 - - -Fe XX 9378 70 - - - 010Fe XVIII 9393 685 - - 074 085Fe X 9401 605 063 072 005 -Fe VIII 9347 56 004 - - -Fe VIII 9362 56 005 - - -Cont 011 012 017 -

131 Aring O VI 12987 545 004 005 - -Fe XXIII 13291 715 - - - 007Fe XXI 12875 705 - - - 083Fe VIII 13094 56 030 025 009 -Fe VIII 13124 56 039 033 013 -Cont 011 020 054 004

171 Aring Ni XIV 17137 635 - - 004 -Fe X 17453 605 - 003 - -Fe IX 17107 585 095 092 080 054Cont - - - 023

193 Aring O V 19290 535 003 - - -Ca XVII 19285 675 - - - 008Ca XIV 19387 655 - - 004 -Fe XXIV 19203 725 - - - 081Fe XII 19512 62 008 018 017 -Fe XII 19351 62 009 019 017 -Fe XII 19239 62 004 009 008 -Fe XI 18823 615 009 010 004 -Fe XI 19283 615 005 006 - -Fe XI 18830 615 004 004 - -Fe X 19004 605 006 004 - -Fe IX 18994 585 006 - - -Fe IX 18850 585 007 - - -Cont - - 005 004

211 Aring Cr IX 21061 595 007 - - -Ca XVI 20860 67 - - - 009Fe XVII 20467 66 - - - 007Fe XIV 21132 63 - 013 039 012Fe XIII 20204 625 - 005 - -Fe XIII 20383 625 - - 007 -Fe XIII 20962 625 - 005 005 -Fe XI 20978 615 011 012 - -Fe X 20745 605 005 003 - -Ni XI 20792 61 003 - - -Cont 008 004 007 041

304 Aring He II 303786 47 033 032 027 029He II 303781 47 066 065 054 058Ca XVIII 30219 685 - - - 005Si XI 30333 62 - - 011 -Cont - - - -

335 Aring Al X 33279 61 005 011 - -Mg VIII 33523 59 011 006 - -Mg VIII 33898 59 011 006 - -Si IX 34195 605 003 003 - -Si VIII 31984 595 004 - - -Fe XVI 33541 645 - - 086 081Fe XIV 33418 63 - 004 004 -Fe X 18454 605 013 015 - -Cont 008 005 - 006

The count rates are normalised for each channel Coronal hole(CH) quiet Sun (QS) active region (AR) and flare (FL) plasma

a Tp corresponds to the log of the temperature of maximum abun-dance

2

Brendan OrsquoDwyer et al SDOAIA response to coronal hole quiet Sun active region and flare plasma

Fig 1 Plot of DEM curves for coronal hole quiet Sun active regionand flare plasma (see text for references)

and AR spectra Density values of Ne = 2 times 108 cmminus3Ne = 5 times 108 cmminus3 and Ne = 5 times 109 cmminus3 were used in cal-culating the CH QS and AR spectra respectively These are thesame density values as those used to calculate the contributionfunctions for the lines which were used to constrain the DEMcurves for the CH QS and AR cases For the flare spectruma value of Ne = 1 times 1011 cmminus3 and the solar coronal abun-dances of Feldman et al (1992) were used The use of eitherphotospheric or coronal abundances in calculating the syntheticspectra reflects the original use of either photospheric or coronalabundances in generating the DEM curves These synthetic spec-tra were then convolved with the effective area of each channelThe effective areas were obtained from P Boerner (2009 privatecommunication) with the exception of the 171 Aring and 335 Aringchannels for which updated versions were used obtained fromSolarsoft (12 July 2010)

3 ResultsTable 1 lists those spectral lines which contribute more than 3to the total emission in each channel for CH QS AR and flareplasma Also included is the fractional contribution of the con-tinuum emission for any case where the continuum contributesmore than 3 to the total emission in a channel Synthetic spec-tra for each of the channels are displayed in Fig 2 - 8 For everychannel each spectrum has been divided by the peak intensityof the strongest spectrum Weaker spectra have been scaled byfactors indicated in each figure

The 94 Aring channel is expected1 to observe the Fe XVIII 9393Aringline [log T[K] sim 685] in flaring regions For both AR and flareplasma (see Fig 2) the dominant contribution comes from theFe XVIII 9393 Aring line However for the CH and QS spectrum (seeFig 2) the dominant contribution comes from the Fe X 9401 Aringline [log T sim 605]

In flaring regions the 131 Aring channel is expected1 to observethe Fe XX 13284 Aring and Fe XXIII 13291 Aring lines However forthe flare spectrum (see Fig 3) the dominant contribution comesfrom the Fe XXI 12875 Aring line [log T sim 705] The combined con-tribution of the Fe XX 13284 Aring and Fe XXIII 13291 Aring lines is lessthan ten percent of the total emission For CH observations the131 Aring channel is expected1 to be dominated by Fe VIII lines [logT sim 56] From our simulations the dominant contribution forCH and QS plasma (see Fig 3) does come from Fe VIII lines butwith a significant contribution from continuum emission For the

Table 1 Predicted AIA count rates

Ion λ T ap Fraction of total emissionAring K CH QS AR FL

94 Aring Mg VIII 9407 59 003 - - -Fe XX 9378 70 - - - 010Fe XVIII 9393 685 - - 074 085Fe X 9401 605 063 072 005 -Fe VIII 9347 56 004 - - -Fe VIII 9362 56 005 - - -Cont 011 012 017 -

131 Aring O VI 12987 545 004 005 - -Fe XXIII 13291 715 - - - 007Fe XXI 12875 705 - - - 083Fe VIII 13094 56 030 025 009 -Fe VIII 13124 56 039 033 013 -Cont 011 020 054 004

171 Aring Ni XIV 17137 635 - - 004 -Fe X 17453 605 - 003 - -Fe IX 17107 585 095 092 080 054Cont - - - 023

193 Aring O V 19290 535 003 - - -Ca XVII 19285 675 - - - 008Ca XIV 19387 655 - - 004 -Fe XXIV 19203 725 - - - 081Fe XII 19512 62 008 018 017 -Fe XII 19351 62 009 019 017 -Fe XII 19239 62 004 009 008 -Fe XI 18823 615 009 010 004 -Fe XI 19283 615 005 006 - -Fe XI 18830 615 004 004 - -Fe X 19004 605 006 004 - -Fe IX 18994 585 006 - - -Fe IX 18850 585 007 - - -Cont - - 005 004

211 Aring Cr IX 21061 595 007 - - -Ca XVI 20860 67 - - - 009Fe XVII 20467 66 - - - 007Fe XIV 21132 63 - 013 039 012Fe XIII 20204 625 - 005 - -Fe XIII 20383 625 - - 007 -Fe XIII 20962 625 - 005 005 -Fe XI 20978 615 011 012 - -Fe X 20745 605 005 003 - -Ni XI 20792 61 003 - - -Cont 008 004 007 041

304 Aring He II 303786 47 033 032 027 029He II 303781 47 066 065 054 058Ca XVIII 30219 685 - - - 005Si XI 30333 62 - - 011 -Cont - - - -

335 Aring Al X 33279 61 005 011 - -Mg VIII 33523 59 011 006 - -Mg VIII 33898 59 011 006 - -Si IX 34195 605 003 003 - -Si VIII 31984 595 004 - - -Fe XVI 33541 645 - - 086 081Fe XIV 33418 63 - 004 004 -Fe X 18454 605 013 015 - -Cont 008 005 - 006

The count rates are normalised for each channel Coronal hole(CH) quiet Sun (QS) active region (AR) and flare (FL) plasma

a Tp corresponds to the log of the temperature of maximum abun-dance

2

From OrsquoDwyer et al 2010

13

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

14

The Atmospheric Imaging Assembly (AIA Lemen et al 2012 Boerner et al 2012) instrument onboard NASArsquos Solar Dynamics Observatory (SDO Pesnell et al 2012) is a suite of four normal-incidence reflecting telescopes that image the Sun in seven EUV channels two UV channels and one visible wavelength channel The aim of this and many other studies is to extract thermal information about the Sunrsquos optically thin corona using the EUV observations The calibrated (ie dark-subtracted flat-fielded and exposure time normalized) count rate yi in the i-th EUV channel is related to the thermal distribution of coronal plasma by

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

where Ki(T) is the temperature response function (see next slide)

Statement of the Problem

15

Let the temperature range be divided into n neighboring bins so that where the j-th temperature bin has range T isin [TjTj+∆Tj) Assuming that Ki(T) is piecewise constant in each temperature bin we have

where DEM(T) is the differential emission measure (in units of cm-5 K-1) of plasma along the line-of-sight ne(T) is the electron number density of plasma at temperature T The challenge is to solve for DEM(T) given a set of EUV measurements y

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Statement of the Problem

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =

Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) is thetemperature response function (in units of DN cm5 s1 pixel1) and DEM(T )dT =

R10

n

2

e(T )dz where DEM(T ) iscalled the dicrarrerential emission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electronnumber density of plasma at a certain temperature T Let the temperature range be divided into n neighboring binsso that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi=nX

j=1

KijEMj where (3)

EMj =

Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj+Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem in

emission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs yimodel

i

2

(6)

is minimized Here yiobs is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of 2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signalsIn an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation resides

in an ane subspace of Rn The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 2006

and

16

The above is a matrix equation of the form y = Kx where bull 13K is an m x n response matrix with each row corresponding to the

temperature response function of one AIA channel bull 13y is an m-tuple corresponding of AIA count (rates) and bull 13x is an n-tuple with components EMj The He II line in the 304 Aring channel is not well-modeled by CHIANTI (Warren 2005 ApJ 157 147) so it is usually not used for DEM analysis So m = 6 for AIA Usually we want more than 6 temperature bins For m lt n the matrix equation y = Kx represents an underdetermined system Matrix elements depends on basis functions used for computing the integral

Statement of the Problem y = Kx

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

17

Function to minimize | y - Kx |2 or |(y - Kx)120706|2 Basically minimize difference between observed and predicted counts The benefits of a least-squares approach is that it leads to Euler-Lagrange equations that can be used to seek (global or local) minima For an overdetermined system we know no single model will fit all the data So 120536-squared minimization is ideal However for underdetermined systems such an approach can be subject to the perils of overfitting Usual way to get around this P a r a m e t e r i z a t i o n e g G u e n n o u e t a l ( 2 0 1 2 a b ) xrt_dem_iterative2pro (M Weber in SSW see also Cheng et al 2012) Regularization eg Hannah amp Kontar (2012) Plowman et al (2013)

Usual Approach 120536-squared Minimization

18

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

19

We address the inverse problem using an approach different than chi-squared minimization The set of solutions satisfying the underdetermined matrix equation y = Kx lies in an affine subspace of Rn We pick the solution x within this subspace such that

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ ~x 0 (13)

K~xmax(~y ~ 0) (14)

Cheung et al 2015 The Sparse Solution

The linear program above finds a solution that minimizes the L1-norm of the solution vector (cf Candes 2006) This is not chi-squared minimization If K is the response function sampled by Dirac Delta functions at specific temperatures this is equivalent to minimizing the total EM If other basis functions are used there is no simple corresponding physical interpretation

20

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Brendan OrsquoDwyer et al SDOAIA response to coronal hole quiet Sun active region and flare plasma

Fig 1 Plot of DEM curves for coronal hole quiet Sun active regionand flare plasma (see text for references)

and AR spectra Density values of Ne = 2 times 108 cmminus3Ne = 5 times 108 cmminus3 and Ne = 5 times 109 cmminus3 were used in cal-culating the CH QS and AR spectra respectively These are thesame density values as those used to calculate the contributionfunctions for the lines which were used to constrain the DEMcurves for the CH QS and AR cases For the flare spectruma value of Ne = 1 times 1011 cmminus3 and the solar coronal abun-dances of Feldman et al (1992) were used The use of eitherphotospheric or coronal abundances in calculating the syntheticspectra reflects the original use of either photospheric or coronalabundances in generating the DEM curves These synthetic spec-tra were then convolved with the effective area of each channelThe effective areas were obtained from P Boerner (2009 privatecommunication) with the exception of the 171 Aring and 335 Aringchannels for which updated versions were used obtained fromSolarsoft (12 July 2010)

3 ResultsTable 1 lists those spectral lines which contribute more than 3to the total emission in each channel for CH QS AR and flareplasma Also included is the fractional contribution of the con-tinuum emission for any case where the continuum contributesmore than 3 to the total emission in a channel Synthetic spec-tra for each of the channels are displayed in Fig 2 - 8 For everychannel each spectrum has been divided by the peak intensityof the strongest spectrum Weaker spectra have been scaled byfactors indicated in each figure

The 94 Aring channel is expected1 to observe the Fe XVIII 9393Aringline [log T[K] sim 685] in flaring regions For both AR and flareplasma (see Fig 2) the dominant contribution comes from theFe XVIII 9393 Aring line However for the CH and QS spectrum (seeFig 2) the dominant contribution comes from the Fe X 9401 Aringline [log T sim 605]

In flaring regions the 131 Aring channel is expected1 to observethe Fe XX 13284 Aring and Fe XXIII 13291 Aring lines However forthe flare spectrum (see Fig 3) the dominant contribution comesfrom the Fe XXI 12875 Aring line [log T sim 705] The combined con-tribution of the Fe XX 13284 Aring and Fe XXIII 13291 Aring lines is lessthan ten percent of the total emission For CH observations the131 Aring channel is expected1 to be dominated by Fe VIII lines [logT sim 56] From our simulations the dominant contribution forCH and QS plasma (see Fig 3) does come from Fe VIII lines butwith a significant contribution from continuum emission For the

Table 1 Predicted AIA count rates

Ion λ T ap Fraction of total emissionAring K CH QS AR FL

94 Aring Mg VIII 9407 59 003 - - -Fe XX 9378 70 - - - 010Fe XVIII 9393 685 - - 074 085Fe X 9401 605 063 072 005 -Fe VIII 9347 56 004 - - -Fe VIII 9362 56 005 - - -Cont 011 012 017 -

131 Aring O VI 12987 545 004 005 - -Fe XXIII 13291 715 - - - 007Fe XXI 12875 705 - - - 083Fe VIII 13094 56 030 025 009 -Fe VIII 13124 56 039 033 013 -Cont 011 020 054 004

171 Aring Ni XIV 17137 635 - - 004 -Fe X 17453 605 - 003 - -Fe IX 17107 585 095 092 080 054Cont - - - 023

193 Aring O V 19290 535 003 - - -Ca XVII 19285 675 - - - 008Ca XIV 19387 655 - - 004 -Fe XXIV 19203 725 - - - 081Fe XII 19512 62 008 018 017 -Fe XII 19351 62 009 019 017 -Fe XII 19239 62 004 009 008 -Fe XI 18823 615 009 010 004 -Fe XI 19283 615 005 006 - -Fe XI 18830 615 004 004 - -Fe X 19004 605 006 004 - -Fe IX 18994 585 006 - - -Fe IX 18850 585 007 - - -Cont - - 005 004

211 Aring Cr IX 21061 595 007 - - -Ca XVI 20860 67 - - - 009Fe XVII 20467 66 - - - 007Fe XIV 21132 63 - 013 039 012Fe XIII 20204 625 - 005 - -Fe XIII 20383 625 - - 007 -Fe XIII 20962 625 - 005 005 -Fe XI 20978 615 011 012 - -Fe X 20745 605 005 003 - -Ni XI 20792 61 003 - - -Cont 008 004 007 041

304 Aring He II 303786 47 033 032 027 029He II 303781 47 066 065 054 058Ca XVIII 30219 685 - - - 005Si XI 30333 62 - - 011 -Cont - - - -

335 Aring Al X 33279 61 005 011 - -Mg VIII 33523 59 011 006 - -Mg VIII 33898 59 011 006 - -Si IX 34195 605 003 003 - -Si VIII 31984 595 004 - - -Fe XVI 33541 645 - - 086 081Fe XIV 33418 63 - 004 004 -Fe X 18454 605 013 015 - -Cont 008 005 - 006

The count rates are normalised for each channel Coronal hole(CH) quiet Sun (QS) active region (AR) and flare (FL) plasma

a Tp corresponds to the log of the temperature of maximum abun-dance

2

Brendan OrsquoDwyer et al SDOAIA response to coronal hole quiet Sun active region and flare plasma

Fig 1 Plot of DEM curves for coronal hole quiet Sun active regionand flare plasma (see text for references)

and AR spectra Density values of Ne = 2 times 108 cmminus3Ne = 5 times 108 cmminus3 and Ne = 5 times 109 cmminus3 were used in cal-culating the CH QS and AR spectra respectively These are thesame density values as those used to calculate the contributionfunctions for the lines which were used to constrain the DEMcurves for the CH QS and AR cases For the flare spectruma value of Ne = 1 times 1011 cmminus3 and the solar coronal abun-dances of Feldman et al (1992) were used The use of eitherphotospheric or coronal abundances in calculating the syntheticspectra reflects the original use of either photospheric or coronalabundances in generating the DEM curves These synthetic spec-tra were then convolved with the effective area of each channelThe effective areas were obtained from P Boerner (2009 privatecommunication) with the exception of the 171 Aring and 335 Aringchannels for which updated versions were used obtained fromSolarsoft (12 July 2010)

3 ResultsTable 1 lists those spectral lines which contribute more than 3to the total emission in each channel for CH QS AR and flareplasma Also included is the fractional contribution of the con-tinuum emission for any case where the continuum contributesmore than 3 to the total emission in a channel Synthetic spec-tra for each of the channels are displayed in Fig 2 - 8 For everychannel each spectrum has been divided by the peak intensityof the strongest spectrum Weaker spectra have been scaled byfactors indicated in each figure

The 94 Aring channel is expected1 to observe the Fe XVIII 9393Aringline [log T[K] sim 685] in flaring regions For both AR and flareplasma (see Fig 2) the dominant contribution comes from theFe XVIII 9393 Aring line However for the CH and QS spectrum (seeFig 2) the dominant contribution comes from the Fe X 9401 Aringline [log T sim 605]

In flaring regions the 131 Aring channel is expected1 to observethe Fe XX 13284 Aring and Fe XXIII 13291 Aring lines However forthe flare spectrum (see Fig 3) the dominant contribution comesfrom the Fe XXI 12875 Aring line [log T sim 705] The combined con-tribution of the Fe XX 13284 Aring and Fe XXIII 13291 Aring lines is lessthan ten percent of the total emission For CH observations the131 Aring channel is expected1 to be dominated by Fe VIII lines [logT sim 56] From our simulations the dominant contribution forCH and QS plasma (see Fig 3) does come from Fe VIII lines butwith a significant contribution from continuum emission For the

Table 1 Predicted AIA count rates

Ion λ T ap Fraction of total emissionAring K CH QS AR FL

94 Aring Mg VIII 9407 59 003 - - -Fe XX 9378 70 - - - 010Fe XVIII 9393 685 - - 074 085Fe X 9401 605 063 072 005 -Fe VIII 9347 56 004 - - -Fe VIII 9362 56 005 - - -Cont 011 012 017 -

131 Aring O VI 12987 545 004 005 - -Fe XXIII 13291 715 - - - 007Fe XXI 12875 705 - - - 083Fe VIII 13094 56 030 025 009 -Fe VIII 13124 56 039 033 013 -Cont 011 020 054 004

171 Aring Ni XIV 17137 635 - - 004 -Fe X 17453 605 - 003 - -Fe IX 17107 585 095 092 080 054Cont - - - 023

193 Aring O V 19290 535 003 - - -Ca XVII 19285 675 - - - 008Ca XIV 19387 655 - - 004 -Fe XXIV 19203 725 - - - 081Fe XII 19512 62 008 018 017 -Fe XII 19351 62 009 019 017 -Fe XII 19239 62 004 009 008 -Fe XI 18823 615 009 010 004 -Fe XI 19283 615 005 006 - -Fe XI 18830 615 004 004 - -Fe X 19004 605 006 004 - -Fe IX 18994 585 006 - - -Fe IX 18850 585 007 - - -Cont - - 005 004

211 Aring Cr IX 21061 595 007 - - -Ca XVI 20860 67 - - - 009Fe XVII 20467 66 - - - 007Fe XIV 21132 63 - 013 039 012Fe XIII 20204 625 - 005 - -Fe XIII 20383 625 - - 007 -Fe XIII 20962 625 - 005 005 -Fe XI 20978 615 011 012 - -Fe X 20745 605 005 003 - -Ni XI 20792 61 003 - - -Cont 008 004 007 041

304 Aring He II 303786 47 033 032 027 029He II 303781 47 066 065 054 058Ca XVIII 30219 685 - - - 005Si XI 30333 62 - - 011 -Cont - - - -

335 Aring Al X 33279 61 005 011 - -Mg VIII 33523 59 011 006 - -Mg VIII 33898 59 011 006 - -Si IX 34195 605 003 003 - -Si VIII 31984 595 004 - - -Fe XVI 33541 645 - - 086 081Fe XIV 33418 63 - 004 004 -Fe X 18454 605 013 015 - -Cont 008 005 - 006

The count rates are normalised for each channel Coronal hole(CH) quiet Sun (QS) active region (AR) and flare (FL) plasma

a Tp corresponds to the log of the temperature of maximum abun-dance

2

From OrsquoDwyer et al 2010

13

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

14

The Atmospheric Imaging Assembly (AIA Lemen et al 2012 Boerner et al 2012) instrument onboard NASArsquos Solar Dynamics Observatory (SDO Pesnell et al 2012) is a suite of four normal-incidence reflecting telescopes that image the Sun in seven EUV channels two UV channels and one visible wavelength channel The aim of this and many other studies is to extract thermal information about the Sunrsquos optically thin corona using the EUV observations The calibrated (ie dark-subtracted flat-fielded and exposure time normalized) count rate yi in the i-th EUV channel is related to the thermal distribution of coronal plasma by

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

where Ki(T) is the temperature response function (see next slide)

Statement of the Problem

15

Let the temperature range be divided into n neighboring bins so that where the j-th temperature bin has range T isin [TjTj+∆Tj) Assuming that Ki(T) is piecewise constant in each temperature bin we have

where DEM(T) is the differential emission measure (in units of cm-5 K-1) of plasma along the line-of-sight ne(T) is the electron number density of plasma at temperature T The challenge is to solve for DEM(T) given a set of EUV measurements y

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Statement of the Problem

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =

Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) is thetemperature response function (in units of DN cm5 s1 pixel1) and DEM(T )dT =

R10

n

2

e(T )dz where DEM(T ) iscalled the dicrarrerential emission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electronnumber density of plasma at a certain temperature T Let the temperature range be divided into n neighboring binsso that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi=nX

j=1

KijEMj where (3)

EMj =

Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj+Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem in

emission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs yimodel

i

2

(6)

is minimized Here yiobs is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of 2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signalsIn an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation resides

in an ane subspace of Rn The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 2006

and

16

The above is a matrix equation of the form y = Kx where bull 13K is an m x n response matrix with each row corresponding to the

temperature response function of one AIA channel bull 13y is an m-tuple corresponding of AIA count (rates) and bull 13x is an n-tuple with components EMj The He II line in the 304 Aring channel is not well-modeled by CHIANTI (Warren 2005 ApJ 157 147) so it is usually not used for DEM analysis So m = 6 for AIA Usually we want more than 6 temperature bins For m lt n the matrix equation y = Kx represents an underdetermined system Matrix elements depends on basis functions used for computing the integral

Statement of the Problem y = Kx

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

17

Function to minimize | y - Kx |2 or |(y - Kx)120706|2 Basically minimize difference between observed and predicted counts The benefits of a least-squares approach is that it leads to Euler-Lagrange equations that can be used to seek (global or local) minima For an overdetermined system we know no single model will fit all the data So 120536-squared minimization is ideal However for underdetermined systems such an approach can be subject to the perils of overfitting Usual way to get around this P a r a m e t e r i z a t i o n e g G u e n n o u e t a l ( 2 0 1 2 a b ) xrt_dem_iterative2pro (M Weber in SSW see also Cheng et al 2012) Regularization eg Hannah amp Kontar (2012) Plowman et al (2013)

Usual Approach 120536-squared Minimization

18

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

19

We address the inverse problem using an approach different than chi-squared minimization The set of solutions satisfying the underdetermined matrix equation y = Kx lies in an affine subspace of Rn We pick the solution x within this subspace such that

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ ~x 0 (13)

K~xmax(~y ~ 0) (14)

Cheung et al 2015 The Sparse Solution

The linear program above finds a solution that minimizes the L1-norm of the solution vector (cf Candes 2006) This is not chi-squared minimization If K is the response function sampled by Dirac Delta functions at specific temperatures this is equivalent to minimizing the total EM If other basis functions are used there is no simple corresponding physical interpretation

20

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

14

The Atmospheric Imaging Assembly (AIA Lemen et al 2012 Boerner et al 2012) instrument onboard NASArsquos Solar Dynamics Observatory (SDO Pesnell et al 2012) is a suite of four normal-incidence reflecting telescopes that image the Sun in seven EUV channels two UV channels and one visible wavelength channel The aim of this and many other studies is to extract thermal information about the Sunrsquos optically thin corona using the EUV observations The calibrated (ie dark-subtracted flat-fielded and exposure time normalized) count rate yi in the i-th EUV channel is related to the thermal distribution of coronal plasma by

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

where Ki(T) is the temperature response function (see next slide)

Statement of the Problem

15

Let the temperature range be divided into n neighboring bins so that where the j-th temperature bin has range T isin [TjTj+∆Tj) Assuming that Ki(T) is piecewise constant in each temperature bin we have

where DEM(T) is the differential emission measure (in units of cm-5 K-1) of plasma along the line-of-sight ne(T) is the electron number density of plasma at temperature T The challenge is to solve for DEM(T) given a set of EUV measurements y

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Statement of the Problem

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =

Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) is thetemperature response function (in units of DN cm5 s1 pixel1) and DEM(T )dT =

R10

n

2

e(T )dz where DEM(T ) iscalled the dicrarrerential emission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electronnumber density of plasma at a certain temperature T Let the temperature range be divided into n neighboring binsso that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi=nX

j=1

KijEMj where (3)

EMj =

Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj+Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem in

emission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs yimodel

i

2

(6)

is minimized Here yiobs is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of 2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signalsIn an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation resides

in an ane subspace of Rn The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 2006

and

16

The above is a matrix equation of the form y = Kx where bull 13K is an m x n response matrix with each row corresponding to the

temperature response function of one AIA channel bull 13y is an m-tuple corresponding of AIA count (rates) and bull 13x is an n-tuple with components EMj The He II line in the 304 Aring channel is not well-modeled by CHIANTI (Warren 2005 ApJ 157 147) so it is usually not used for DEM analysis So m = 6 for AIA Usually we want more than 6 temperature bins For m lt n the matrix equation y = Kx represents an underdetermined system Matrix elements depends on basis functions used for computing the integral

Statement of the Problem y = Kx

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

17

Function to minimize | y - Kx |2 or |(y - Kx)120706|2 Basically minimize difference between observed and predicted counts The benefits of a least-squares approach is that it leads to Euler-Lagrange equations that can be used to seek (global or local) minima For an overdetermined system we know no single model will fit all the data So 120536-squared minimization is ideal However for underdetermined systems such an approach can be subject to the perils of overfitting Usual way to get around this P a r a m e t e r i z a t i o n e g G u e n n o u e t a l ( 2 0 1 2 a b ) xrt_dem_iterative2pro (M Weber in SSW see also Cheng et al 2012) Regularization eg Hannah amp Kontar (2012) Plowman et al (2013)

Usual Approach 120536-squared Minimization

18

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

19

We address the inverse problem using an approach different than chi-squared minimization The set of solutions satisfying the underdetermined matrix equation y = Kx lies in an affine subspace of Rn We pick the solution x within this subspace such that

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ ~x 0 (13)

K~xmax(~y ~ 0) (14)

Cheung et al 2015 The Sparse Solution

The linear program above finds a solution that minimizes the L1-norm of the solution vector (cf Candes 2006) This is not chi-squared minimization If K is the response function sampled by Dirac Delta functions at specific temperatures this is equivalent to minimizing the total EM If other basis functions are used there is no simple corresponding physical interpretation

20

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

The Atmospheric Imaging Assembly (AIA Lemen et al 2012 Boerner et al 2012) instrument onboard NASArsquos Solar Dynamics Observatory (SDO Pesnell et al 2012) is a suite of four normal-incidence reflecting telescopes that image the Sun in seven EUV channels two UV channels and one visible wavelength channel The aim of this and many other studies is to extract thermal information about the Sunrsquos optically thin corona using the EUV observations The calibrated (ie dark-subtracted flat-fielded and exposure time normalized) count rate yi in the i-th EUV channel is related to the thermal distribution of coronal plasma by

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

where Ki(T) is the temperature response function (see next slide)

Statement of the Problem

15

Let the temperature range be divided into n neighboring bins so that where the j-th temperature bin has range T isin [TjTj+∆Tj) Assuming that Ki(T) is piecewise constant in each temperature bin we have

where DEM(T) is the differential emission measure (in units of cm-5 K-1) of plasma along the line-of-sight ne(T) is the electron number density of plasma at temperature T The challenge is to solve for DEM(T) given a set of EUV measurements y

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Statement of the Problem

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =

Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) is thetemperature response function (in units of DN cm5 s1 pixel1) and DEM(T )dT =

R10

n

2

e(T )dz where DEM(T ) iscalled the dicrarrerential emission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electronnumber density of plasma at a certain temperature T Let the temperature range be divided into n neighboring binsso that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi=nX

j=1

KijEMj where (3)

EMj =

Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj+Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem in

emission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs yimodel

i

2

(6)

is minimized Here yiobs is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of 2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signalsIn an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation resides

in an ane subspace of Rn The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 2006

and

16

The above is a matrix equation of the form y = Kx where bull 13K is an m x n response matrix with each row corresponding to the

temperature response function of one AIA channel bull 13y is an m-tuple corresponding of AIA count (rates) and bull 13x is an n-tuple with components EMj The He II line in the 304 Aring channel is not well-modeled by CHIANTI (Warren 2005 ApJ 157 147) so it is usually not used for DEM analysis So m = 6 for AIA Usually we want more than 6 temperature bins For m lt n the matrix equation y = Kx represents an underdetermined system Matrix elements depends on basis functions used for computing the integral

Statement of the Problem y = Kx

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

17

Function to minimize | y - Kx |2 or |(y - Kx)120706|2 Basically minimize difference between observed and predicted counts The benefits of a least-squares approach is that it leads to Euler-Lagrange equations that can be used to seek (global or local) minima For an overdetermined system we know no single model will fit all the data So 120536-squared minimization is ideal However for underdetermined systems such an approach can be subject to the perils of overfitting Usual way to get around this P a r a m e t e r i z a t i o n e g G u e n n o u e t a l ( 2 0 1 2 a b ) xrt_dem_iterative2pro (M Weber in SSW see also Cheng et al 2012) Regularization eg Hannah amp Kontar (2012) Plowman et al (2013)

Usual Approach 120536-squared Minimization

18

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

19

We address the inverse problem using an approach different than chi-squared minimization The set of solutions satisfying the underdetermined matrix equation y = Kx lies in an affine subspace of Rn We pick the solution x within this subspace such that

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ ~x 0 (13)

K~xmax(~y ~ 0) (14)

Cheung et al 2015 The Sparse Solution

The linear program above finds a solution that minimizes the L1-norm of the solution vector (cf Candes 2006) This is not chi-squared minimization If K is the response function sampled by Dirac Delta functions at specific temperatures this is equivalent to minimizing the total EM If other basis functions are used there is no simple corresponding physical interpretation

20

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Let the temperature range be divided into n neighboring bins so that where the j-th temperature bin has range T isin [TjTj+∆Tj) Assuming that Ki(T) is piecewise constant in each temperature bin we have

where DEM(T) is the differential emission measure (in units of cm-5 K-1) of plasma along the line-of-sight ne(T) is the electron number density of plasma at temperature T The challenge is to solve for DEM(T) given a set of EUV measurements y

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Statement of the Problem

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =

Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) is thetemperature response function (in units of DN cm5 s1 pixel1) and DEM(T )dT =

R10

n

2

e(T )dz where DEM(T ) iscalled the dicrarrerential emission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electronnumber density of plasma at a certain temperature T Let the temperature range be divided into n neighboring binsso that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi=nX

j=1

KijEMj where (3)

EMj =

Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj+Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem in

emission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs yimodel

i

2

(6)

is minimized Here yiobs is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of 2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signalsIn an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation resides

in an ane subspace of Rn The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 2006

and

16

The above is a matrix equation of the form y = Kx where bull 13K is an m x n response matrix with each row corresponding to the

temperature response function of one AIA channel bull 13y is an m-tuple corresponding of AIA count (rates) and bull 13x is an n-tuple with components EMj The He II line in the 304 Aring channel is not well-modeled by CHIANTI (Warren 2005 ApJ 157 147) so it is usually not used for DEM analysis So m = 6 for AIA Usually we want more than 6 temperature bins For m lt n the matrix equation y = Kx represents an underdetermined system Matrix elements depends on basis functions used for computing the integral

Statement of the Problem y = Kx

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

17

Function to minimize | y - Kx |2 or |(y - Kx)120706|2 Basically minimize difference between observed and predicted counts The benefits of a least-squares approach is that it leads to Euler-Lagrange equations that can be used to seek (global or local) minima For an overdetermined system we know no single model will fit all the data So 120536-squared minimization is ideal However for underdetermined systems such an approach can be subject to the perils of overfitting Usual way to get around this P a r a m e t e r i z a t i o n e g G u e n n o u e t a l ( 2 0 1 2 a b ) xrt_dem_iterative2pro (M Weber in SSW see also Cheng et al 2012) Regularization eg Hannah amp Kontar (2012) Plowman et al (2013)

Usual Approach 120536-squared Minimization

18

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

19

We address the inverse problem using an approach different than chi-squared minimization The set of solutions satisfying the underdetermined matrix equation y = Kx lies in an affine subspace of Rn We pick the solution x within this subspace such that

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ ~x 0 (13)

K~xmax(~y ~ 0) (14)

Cheung et al 2015 The Sparse Solution

The linear program above finds a solution that minimizes the L1-norm of the solution vector (cf Candes 2006) This is not chi-squared minimization If K is the response function sampled by Dirac Delta functions at specific temperatures this is equivalent to minimizing the total EM If other basis functions are used there is no simple corresponding physical interpretation

20

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

The above is a matrix equation of the form y = Kx where bull 13K is an m x n response matrix with each row corresponding to the

temperature response function of one AIA channel bull 13y is an m-tuple corresponding of AIA count (rates) and bull 13x is an n-tuple with components EMj The He II line in the 304 Aring channel is not well-modeled by CHIANTI (Warren 2005 ApJ 157 147) so it is usually not used for DEM analysis So m = 6 for AIA Usually we want more than 6 temperature bins For m lt n the matrix equation y = Kx represents an underdetermined system Matrix elements depends on basis functions used for computing the integral

Statement of the Problem y = Kx

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

Coronal Mass Source for Post-Eruption Arcades 5

4 EMISSION MEASURE DETERMINATION USING AIA

41 Statement of the problem

AIA EUV observations can be related to the physical properties of optically thin coronal plasma as an integral overtemperature space

yi =Z 1

0

Ki(T ) DEM(T )dT (1)

where yi is the exposure time-normalized pixel value in the i-th AIA channel (in units of DN s1 pixel1) Ki(T ) isthe temperature response function (in units of DN cm5 s1 pixel1) and DEM(T ) =

R10

n

2

e(T )dz is the dicrarrerentialemission measure (in units of cm5 K1) of the plasma along a line-of-sight ne(T ) is the electron number density ofplasma at a certain temperature T Let the temperature range be divided into n neighboring bins so that

yi =nX

j=1

Z Tj+Tj

Tj

Ki(T )DEM(T )dT (2)

where the j-th temperature bin has range T 2 [Tj Tj + Tj) The aim is to use AIA measurements to retrieve theemission measure distribution (either in dicrarrerential or integral form)

Assume that Ki(T ) = Kij is piecewise constant in each j-th temperature bin so

yi =nX

j=1

KijEMj where (3)

EMj =Z Tj+Tj

Tj

DEM(T )dT (4)

With a priori knowledge about the form of DEM(T ) over the range [Tj Tj +Tj) it is in principle possible to drop theassumption that Ki(T ) be piecewise constant and define a DEM-weighted temperature response function Howeverwe adopt this assumption since such information is not availableEq (3) can be written in the following form

y = Kx (5)

where K is an m n matrix with components Kij y is an m-tuple corresponding to measurements by the AIA EUVchannels (m = 6 when using the 94 131 171 193 211 and 335 A channels) and x is an n-tuple with componentsgiven by EMj

For m lt n (ie more than 6 temperature bins) Eq (5) is underdetermined This is a well-known problem inemission measure inversions and thus far researchers have attempted to tackle this problem using a least-squaresapproach That is the DEM solution is chosen to be one such that

2 =mX

i=1

yiobs

yimodel

i

2

(6)

is minimized Here yiobs

is the observed intensity value for i-th AIA channel yimodel

is the predicted value for a given(D)EM model and i is the uncertainty for the i-th channel The benefit of this type of least-squares approach is thatit results in Euler-Lagrange equations that can be used to seek (global or local) minima This approach is ideal inoverdetermined systems (ie m gt n) where it is known that no single model will reproduce all n measurements (linearregression through three or more non-colinear points is one example) However for underdetermined systems suchan approach is subject to the perils of overfitting To mitigate this regularization terms are sometimes added to thedefinition of

2 to impose additional constraints such as smoothness in the solution Other methods impose that theDEM solution have a certain shape (eg a Gaussian or power-law distribution) so that the number of free parametersis less than m

42 Sparse Emission Measure Solution

We address the inverse problem using a dicrarrerent approach We do so by applying lessons learned from the field ofcompressed sensing Compressed sensing is concerned with the recovery of signals where the number of measurementsis (much) less than the number of components in the reconstructed signals

In an underdetermined linear system such as given by Eq (5) the family of solutions satisfying the equation residesin an ane subspace of R

n The challenge is to select a solution within this subspace that most faithfully representsthe underlying scenario In a series of papers on solutions to underdetermined linear systems Candes amp Tao (eg 20062007) showed that the assumption of sparsity often leads to a better solution than a least-squaresminimum energy

17

Function to minimize | y - Kx |2 or |(y - Kx)120706|2 Basically minimize difference between observed and predicted counts The benefits of a least-squares approach is that it leads to Euler-Lagrange equations that can be used to seek (global or local) minima For an overdetermined system we know no single model will fit all the data So 120536-squared minimization is ideal However for underdetermined systems such an approach can be subject to the perils of overfitting Usual way to get around this P a r a m e t e r i z a t i o n e g G u e n n o u e t a l ( 2 0 1 2 a b ) xrt_dem_iterative2pro (M Weber in SSW see also Cheng et al 2012) Regularization eg Hannah amp Kontar (2012) Plowman et al (2013)

Usual Approach 120536-squared Minimization

18

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

19

We address the inverse problem using an approach different than chi-squared minimization The set of solutions satisfying the underdetermined matrix equation y = Kx lies in an affine subspace of Rn We pick the solution x within this subspace such that

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ ~x 0 (13)

K~xmax(~y ~ 0) (14)

Cheung et al 2015 The Sparse Solution

The linear program above finds a solution that minimizes the L1-norm of the solution vector (cf Candes 2006) This is not chi-squared minimization If K is the response function sampled by Dirac Delta functions at specific temperatures this is equivalent to minimizing the total EM If other basis functions are used there is no simple corresponding physical interpretation

20

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Function to minimize | y - Kx |2 or |(y - Kx)120706|2 Basically minimize difference between observed and predicted counts The benefits of a least-squares approach is that it leads to Euler-Lagrange equations that can be used to seek (global or local) minima For an overdetermined system we know no single model will fit all the data So 120536-squared minimization is ideal However for underdetermined systems such an approach can be subject to the perils of overfitting Usual way to get around this P a r a m e t e r i z a t i o n e g G u e n n o u e t a l ( 2 0 1 2 a b ) xrt_dem_iterative2pro (M Weber in SSW see also Cheng et al 2012) Regularization eg Hannah amp Kontar (2012) Plowman et al (2013)

Usual Approach 120536-squared Minimization

18

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

19

We address the inverse problem using an approach different than chi-squared minimization The set of solutions satisfying the underdetermined matrix equation y = Kx lies in an affine subspace of Rn We pick the solution x within this subspace such that

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ ~x 0 (13)

K~xmax(~y ~ 0) (14)

Cheung et al 2015 The Sparse Solution

The linear program above finds a solution that minimizes the L1-norm of the solution vector (cf Candes 2006) This is not chi-squared minimization If K is the response function sampled by Dirac Delta functions at specific temperatures this is equivalent to minimizing the total EM If other basis functions are used there is no simple corresponding physical interpretation

20

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

19

We address the inverse problem using an approach different than chi-squared minimization The set of solutions satisfying the underdetermined matrix equation y = Kx lies in an affine subspace of Rn We pick the solution x within this subspace such that

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ ~x 0 (13)

K~xmax(~y ~ 0) (14)

Cheung et al 2015 The Sparse Solution

The linear program above finds a solution that minimizes the L1-norm of the solution vector (cf Candes 2006) This is not chi-squared minimization If K is the response function sampled by Dirac Delta functions at specific temperatures this is equivalent to minimizing the total EM If other basis functions are used there is no simple corresponding physical interpretation

20

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

We address the inverse problem using an approach different than chi-squared minimization The set of solutions satisfying the underdetermined matrix equation y = Kx lies in an affine subspace of Rn We pick the solution x within this subspace such that

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ ~x 0 (13)

K~xmax(~y ~ 0) (14)

Cheung et al 2015 The Sparse Solution

The linear program above finds a solution that minimizes the L1-norm of the solution vector (cf Candes 2006) This is not chi-squared minimization If K is the response function sampled by Dirac Delta functions at specific temperatures this is equivalent to minimizing the total EM If other basis functions are used there is no simple corresponding physical interpretation

20

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

We are unaware of physical principles pertaining to coronal plasma that motivate the optimization problem posed above However this choice has some important benefits 1)It does not overfit (consistent with the principle of

parsimony ie Ockhamrsquos Razor) 2)It ensures positivity of the solution (if solutions

exist) 3)It is an L1-norm minimization problem so we can use

standard techniques from compressed sensing (cf Candes amp Tao 2006)

13 BTW the L1-norm of a vector x = 120680 |xi| 4) Speed O(104) solutions sec with single IDL thread

The Sparse Solution

21

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

In practice measurement uncertainties imply that the equality y = Kx may not be satisfied So our method solves the followed modified linear program

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

6

approach To be specific the most sparse solution is one that

minimizes k ~x kl0 subject to K~x = ~y (7)

Here k ~x kl0 is the l-zero norm of ~x which is just the number of non-zero components of ~x Since there is no ecientalgorithm for solving this l-zero minimization problem Candes amp Tao (2006) instead proposed that one should solvethe corresponding l-1 minimization problem namely

minimize k ~x kl1 subject to K~x = ~y (8)

where k ~x kl1=nP

j=1

k xj k This is the underpinning of our approach to tackling the EM inversion problem Since

the objective function is convex algorithms developed for convex optimization can be used to solve for ~x In practiceuncertainties in the instrument response matrix (Kij) as well as in the measurements (~y) means that the sought-aftersolution may not necessarily satisfy Eq (5) Furthermore for EMs we must impose that the solution be positivesemidefinite (ie EMj 0) So our method solves the following linear program

minimize k ~x kl1 subject to K~x~y + ~ (9)

K~xmax(~y ~ 0) (10)

~x 0 (11)

The inequality conditions (13) and (14) provide some tolerance for the solution to deviate from satisfying Eq (5) Forthe implementation we choose j = 2

pyj Other than the problem as stated by (13) - (14) no other conditions (eg

the shape of the EM curve) are imposed

minimizenX

j

xj subject to K~x = ~y ~x 0 (12)

minimizenX

j

xj subject to K~x~y + ~ (13)

~x 0 K~xmax(~y ~ 0) (14)

The vector η is a measure of the uncertainty in the count rate and provides tolerance for the predicted counts (Kx) to deviate from the observed values (y) To enforce positive counts the lower bound is set to max(y-η 0)

Handling noise

22

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

94

131

171

193

211

335

Crossmarks are observed counts

94

131

171

193

211

335

Sparse inversion120536-squared minimization

Subspace of allowed solutions in our method

vs

23

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Basis Functions for DEMThermal Diagnostics with SDOAIA A Validated Method for DEMs 17

APPENDIX

QUADRATURE SCHEME

Let i = 1 2 m denote the index over a set of wavelength band channels andor line spectra Let the DEM functionbe written in terms of a set of positive semidefinite basis functions bj(log T ) 0 | k = 1 2 l viz

DEM(log T ) =lX

k=1

bk(log T )xk (A1)

with quadrature coecients xk 0 Approximating the integrals in equation (1) as sums in log T space we have

yi =nX

j=1

lX

k=1

KijBjkxk log T (A2)

where j = 1 2 n is the index over temperature bins Kij = Ki(log Tj) and Bjk = bk(log Tj) The response matrixK = (Kij) has dimensions m n The basis matrix B = (Bjk) has dimensions n l with the k-th column vectorcorresponding to the k-th basis function bk(log Tj) Defining the dictionary matrix D = KB the set of integralequations (1) can be written in matrix form

~y = D~x (A3)

where the sought-after solution vector ~x is an l-tuple with components xk log T (k = 1 2 l) When the numberof basis functions exceeds the number of image channels (ie l gt m) the linear system Eq (A3) is underdeterminedFor the results in this paper we use an equidistant grid in log T with log T = 01 ranging from log T = 55 to 75

(ie n = 21) Over this temperature grid the set of Dirac-delta basis functions bDirac

k | k = 1 n is

b

Dirac

k (log Tj)=1 if log Tj = log Tk (A4)

=0 otherwise (A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions So for the set of Dirac-deltafunctions BDirac = I (the identity matrix)In addition to Dirac-delta functions we also use basis functions consisting of truncated Gaussians Each Gaussian

function of width a generates a set of basis functions bak | k = 1 n where

b

a

k(log Tj)=exp

(log Tj log Tk)2

a

2

if | log Tj log Tk| 18a (A6)

=0 otherwise (A7)

The Gaussian basis functions are truncated (ie set to zero) for values of log Tj outside the temperature grid usedfor inversions The corresponding basis matrix for this set is denoted Ba Dicrarrerent sets of basis functions can becombined by concatenating their associated basis matrices For the inversions shown here we use the combined basismatrix

B =BDirac|Ba=01|Ba=02|Ba=06

(A8)

Note the individual Gaussian basis functions are not normalized by their sums (ie all have maximum value of unity attheir peaks) So given multiple solutions that equally fit the data the method will prefer a solution consisting of a singlebroad Gaussian over solutions consisting of multiple narrow Gaussians (andor Dirac-delta functions) Empiricallywe find the choice of not normalizing the Gaussian basis functions results in better inversions results (more on thisbelow) Because n = 21 B as indicated above (see Fig 15 for a graphical representation) has dimensions 21 84and D has dimensions m 84 With six AIA channels m = 6 Even when AIA is augmented by XRT or EIS datam 84 This makes Eq (A3) a highly underdetermined system which we solve by the method of basis pursuit (seesection 3)Seeking to solve this underdetermined system is the same as the following geometric problem Suppose we aim to

express some given column vector ~y (of dimension m) as the linear combination of members drawn from a family of

column vectors Let this family of column vectors be denoted ~dk | k = 1 l The goal is to find coecients xk

such that ~y =Pl

k=1

xk~

dk which is equivalent to the linear system (A3) if the dictionary matrix D is constructed by

concatenating the ~

dkrsquos side by side Because l gt m ~dk | k = 1 l is an overcomplete set of possible basis vectors(ie dictionary) for building up ~y So the non-uniqueness of a DEM solution satisfying Eq (A3) is the same as themultiplicity of ways to find a basis for ~y Basis pursuit addresses this by seeking a solution that minimizes the L1-norm|~x|

1

In other words basis pursuit finds the most sparse representation of ~y from an overcomplete dictionary D (Chenet al 1998)

Tem

pera

ture

Bin

s

In practice we solve this

24

Basis matrix B

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

94 DNs131 DNs171 DNs193 DNs211 DNs335 DNs

y

=

D = KB xGeometrical Interpretation of Problem

We seek to build a column vector y by linear combinations of columns of the matrix D=KB xj are the coefficients of the linear combination More columns of KB from which to chose than components of y =gt underdetermined problemDo ldquobasis pursuitrdquo by minimizing L1 norm of x

25

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Application to real AIA data

26

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Warren Brooks amp Winebarger (2011)

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Side benefit Image Denoising

y (AIA 131 Level 15) Dx (from inversion)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

30

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

How to use the Sparse DEM code

bull Instructions in the readme file

bull Download genx files which contain sample AIA 6-channel data

bull Download aia_sparse_em_initpro amp exercise1pro

compile aia_sparse_em_init

r exercise1 Example of DEM inversion

r exercise2 Example of DEM sensitivity to noise

httptinyurlcomaiadem

31

Author Mark Cheung Purpose Sample script for running sparse DEM inversion (see Cheung et al 2015) for details Revision history 2015-10-20 First version 2015-10-23 Added note about lgT axis files = file_list(genx) aiadatafile = files[0] Restore some prepared AIA level 15 data (ie already been aia_preped) restgen file=aiadatafile struct=s Initialize solver This step builds up the response functions (which are time-dependent) and the basis functions If running inversions over a set of data spanning over a few hours or even days it is not necessary to re-initialize IF (0) THEN BEGIN As discussed in the Appendix of Cheung et al (2015) the inversion can push some EM into temperature bins if the lgT axis goes below logT ~ 57 (see Fig 18) It is suggested the user check the dependence of the solution by varying lgTmin lgTmin = 57 minimum for lgT axis for inversion dlgT = 01 width of lgT bin nlgT = 21 number of lgT bins aia_sparse_em_init timedepend = soindex[0]date_obs evenorm $ use_lgtaxis=findgen(nlgT)dlgT+lgTminlgtaxis = aia_sparse_em_lgtaxis() ENDIF

We will use the data in simg to invert simg is a stack of level 15 exposure time normalized AIA pixel values exptimestr = [+strjoin(string(soindexexptimeformat=(F85)))+] This defines the tolerance function for the inversion y denotes the values in DNspixel (ie level 15 exposure time normalized) aia_bp_estimate_error gives the estimated uncertainty for a given DN pixel for a given AIA channel Since y contains DNpixels we have to multiply by exposure time first to pass aia_bp_estimate_error Then we will divide the output of aia_bp_estimate_error by the exposure time again If one wants to include uncertainties in atomic data suggest to add the temp keyword to aia_bp_estimate_error() tolfunc = aia_bp_estimate_error(y+exptimestr+ [94131171193211335] $ num_images=lsquo+strtrim(string(sbinning^2format=(I4))2)+)+exptimestr Do DEM solve Note The solver assumes the third dimension is arranged according to [94131171193211335] aia_sparse_em_solve simg tolfunc=tolfunc tolfac=14 oem=emcube status=status coeff=coeff

Output of exercise1pro

status[Nx Ny] - Indicating where a DEM solution was found 0 is yes coeff[Nx Ny 84] - Coefficients of basis functions used to express DEM ie the x in equation y = KBx emcube[Nx Ny 21] - DEM solution ie Bx 34

Interactively inspect DEM profilesbull aia_sparse_dem_inspect coeff emcube status ylog

35

36

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

37

Active Region Thermal Structure

ldquofor this value of f [=03] the coefficients to the polynomial fit are minus731times10minus2 975 times 10minus1 990times10minus2 and minus284times10minus3rdquo

Warren Winebarger amp Brooks (2012) devised a method to separate the lsquowarmrsquo and lsquohotrsquo components of emission from the AIA 94 Aring channel They use this empirical fit

38

39

AR 11190Workshop Practice Exercise 1

Go to httpwwwlmsalcom~cheungAIAar11190 Download a genx file and plot the DEM distribution in regions marked by the green boxes How do the DEMs in these boxes evolve What about the DEM averaged over the AR (Take care with pixels that have no DEM solutions)

40

From Warren Winebarger amp Brooks (2012)

We will use the data in simg to invert simg is a stack of level 15 exposure time normalized AIA pixel values exptimestr = [+strjoin(string(soindexexptimeformat=(F85)))+] This defines the tolerance function for the inversion y denotes the values in DNspixel (ie level 15 exposure time normalized) aia_bp_estimate_error gives the estimated uncertainty for a given DN pixel for a given AIA channel Since y contains DNpixels we have to multiply by exposure time first to pass aia_bp_estimate_error Then we will divide the output of aia_bp_estimate_error by the exposure time again If one wants to include uncertainties in atomic data suggest to add the temp keyword to aia_bp_estimate_error() tolfunc = aia_bp_estimate_error(y+exptimestr+ [94131171193211335] $ num_images=lsquo+strtrim(string(sbinning^2format=(I4))2)+)+exptimestr Do DEM solve Note The solver assumes the third dimension is arranged according to [94131171193211335] aia_sparse_em_solve simg tolfunc=tolfunc tolfac=14 oem=emcube status=status coeff=coeff

Output of exercise1pro

status[Nx Ny] - Indicating where a DEM solution was found 0 is yes coeff[Nx Ny 84] - Coefficients of basis functions used to express DEM ie the x in equation y = KBx emcube[Nx Ny 21] - DEM solution ie Bx 34

Interactively inspect DEM profilesbull aia_sparse_dem_inspect coeff emcube status ylog

35

36

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

37

Active Region Thermal Structure

ldquofor this value of f [=03] the coefficients to the polynomial fit are minus731times10minus2 975 times 10minus1 990times10minus2 and minus284times10minus3rdquo

Warren Winebarger amp Brooks (2012) devised a method to separate the lsquowarmrsquo and lsquohotrsquo components of emission from the AIA 94 Aring channel They use this empirical fit

38

39

AR 11190Workshop Practice Exercise 1

Go to httpwwwlmsalcom~cheungAIAar11190 Download a genx file and plot the DEM distribution in regions marked by the green boxes How do the DEMs in these boxes evolve What about the DEM averaged over the AR (Take care with pixels that have no DEM solutions)

40

From Warren Winebarger amp Brooks (2012)

Output of exercise1pro

status[Nx Ny] - Indicating where a DEM solution was found 0 is yes coeff[Nx Ny 84] - Coefficients of basis functions used to express DEM ie the x in equation y = KBx emcube[Nx Ny 21] - DEM solution ie Bx 34

Interactively inspect DEM profilesbull aia_sparse_dem_inspect coeff emcube status ylog

35

36

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

37

Active Region Thermal Structure

ldquofor this value of f [=03] the coefficients to the polynomial fit are minus731times10minus2 975 times 10minus1 990times10minus2 and minus284times10minus3rdquo

Warren Winebarger amp Brooks (2012) devised a method to separate the lsquowarmrsquo and lsquohotrsquo components of emission from the AIA 94 Aring channel They use this empirical fit

38

39

AR 11190Workshop Practice Exercise 1

Go to httpwwwlmsalcom~cheungAIAar11190 Download a genx file and plot the DEM distribution in regions marked by the green boxes How do the DEMs in these boxes evolve What about the DEM averaged over the AR (Take care with pixels that have no DEM solutions)

40

From Warren Winebarger amp Brooks (2012)

Interactively inspect DEM profilesbull aia_sparse_dem_inspect coeff emcube status ylog

35

36

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

37

Active Region Thermal Structure

ldquofor this value of f [=03] the coefficients to the polynomial fit are minus731times10minus2 975 times 10minus1 990times10minus2 and minus284times10minus3rdquo

Warren Winebarger amp Brooks (2012) devised a method to separate the lsquowarmrsquo and lsquohotrsquo components of emission from the AIA 94 Aring channel They use this empirical fit

38

39

AR 11190Workshop Practice Exercise 1

Go to httpwwwlmsalcom~cheungAIAar11190 Download a genx file and plot the DEM distribution in regions marked by the green boxes How do the DEMs in these boxes evolve What about the DEM averaged over the AR (Take care with pixels that have no DEM solutions)

40

From Warren Winebarger amp Brooks (2012)

36

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

37

Active Region Thermal Structure

ldquofor this value of f [=03] the coefficients to the polynomial fit are minus731times10minus2 975 times 10minus1 990times10minus2 and minus284times10minus3rdquo

Warren Winebarger amp Brooks (2012) devised a method to separate the lsquowarmrsquo and lsquohotrsquo components of emission from the AIA 94 Aring channel They use this empirical fit

38

39

AR 11190Workshop Practice Exercise 1

Go to httpwwwlmsalcom~cheungAIAar11190 Download a genx file and plot the DEM distribution in regions marked by the green boxes How do the DEMs in these boxes evolve What about the DEM averaged over the AR (Take care with pixels that have no DEM solutions)

40

From Warren Winebarger amp Brooks (2012)

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle

37

Active Region Thermal Structure

ldquofor this value of f [=03] the coefficients to the polynomial fit are minus731times10minus2 975 times 10minus1 990times10minus2 and minus284times10minus3rdquo

Warren Winebarger amp Brooks (2012) devised a method to separate the lsquowarmrsquo and lsquohotrsquo components of emission from the AIA 94 Aring channel They use this empirical fit

38

39

AR 11190Workshop Practice Exercise 1

Go to httpwwwlmsalcom~cheungAIAar11190 Download a genx file and plot the DEM distribution in regions marked by the green boxes How do the DEMs in these boxes evolve What about the DEM averaged over the AR (Take care with pixels that have no DEM solutions)

40

From Warren Winebarger amp Brooks (2012)

Active Region Thermal Structure

ldquofor this value of f [=03] the coefficients to the polynomial fit are minus731times10minus2 975 times 10minus1 990times10minus2 and minus284times10minus3rdquo

Warren Winebarger amp Brooks (2012) devised a method to separate the lsquowarmrsquo and lsquohotrsquo components of emission from the AIA 94 Aring channel They use this empirical fit

38

39

AR 11190Workshop Practice Exercise 1

Go to httpwwwlmsalcom~cheungAIAar11190 Download a genx file and plot the DEM distribution in regions marked by the green boxes How do the DEMs in these boxes evolve What about the DEM averaged over the AR (Take care with pixels that have no DEM solutions)

40

From Warren Winebarger amp Brooks (2012)

39

AR 11190Workshop Practice Exercise 1

Go to httpwwwlmsalcom~cheungAIAar11190 Download a genx file and plot the DEM distribution in regions marked by the green boxes How do the DEMs in these boxes evolve What about the DEM averaged over the AR (Take care with pixels that have no DEM solutions)

40

From Warren Winebarger amp Brooks (2012)

AR 11190Workshop Practice Exercise 1

Go to httpwwwlmsalcom~cheungAIAar11190 Download a genx file and plot the DEM distribution in regions marked by the green boxes How do the DEMs in these boxes evolve What about the DEM averaged over the AR (Take care with pixels that have no DEM solutions)

40

From Warren Winebarger amp Brooks (2012)

AR 11190Workshop Practice Exercise 2

Go to httpwwwlmsalcom~cheungAIA11190 Download a genx file Starting from the DEM solution generate AIA 94 images separately for the lsquowarmrsquo and lsquohotrsquo components Hint use PRO AIA_SPARSE_EM_EM2IMAGE em image=image status=status

41

From Warren Winebarger amp Brooks (2012)

bull By default aia_sparse_em_init uses 4 sets of basis functions (Dirac + 3 sets of truncated gaussians)

bull Default keyword is bases_sigmas = [0010206]

bull Test how choosing other basis sets changes DEM results eg

bull Only Dirac Delta functions bases_sigmas = [0]

bull Dirac + Gaussians of width 01 bases_sigmas = [001]

bull Dirac + Gaussians of width 01 02 and 03 bases_sigmas = [0010203]

Workshop Practice Exercise 3 Examine impact of choice of basis functions

bull Joint AIA-XRT DEM inversions

bull Download aiaxrtpro from httptinyurlcomaiadem

bull compile aia_sparse_em_init

bull r aiaxrt

bull This script solves uses sample AIA data to solve for a DEM cube Then it synthesizes AIA and XRT images Then it uses AIA+XRT for DEM inversion

Workshop Practice Exercise 4 AIA + XRT DEM inversions

bull Joint AIA-XRT DEM inversions

bull Download aiaxrtpro from httptinyurlcomaiadem

bull compile aia_sparse_em_init

bull r aiaxrt

bull This script solves uses sample AIA data to solve for a DEM cube Then it synthesizes AIA and XRT images Then it uses AIA+XRT for DEM inversion

Workshop Practice Exercise 4 AIA + XRT DEM inversions

Outline1 Aims

2 What do we see in the EUV channels of AIA

3 Introduction to the problem of Differential Emission Measure (DEM) Inversions

4 Description of sparse solver for AIA data

5 Tutorial on how to use the AIA_SPARSE_EM package

6 Practice exercises

7 Example science problems to tackle 1315pm today

44

Science Cases

bull Thermal evolution of active regions from emergence to decay

bull Thermodynamic structure of reconnection outflows

bull From chromospheric evaporation to catastrophic condensation

Lower right panel only Greyscale Blos from HMI Green 6MK EM YellowRed 10 MK EM

Other panels EM in various log T bins

DEM movie of the emergence of AR 11726

Science Case 1) Emerging Flux

Black pixels are where no solutions were found

DEM movie of the emergence of AR 11158

The Astrophysical Journal 780107 (6pp) 2014 January 1 Krucker amp Battaglia

0515 0520 0525 0530 0535UT on 2012-Jul-19

0

20

40

60

80

100

120

coun

ts s

-1

RHESSI 30-80 keV

GOESgt3 keV

900 920 940 960 980 1000 1020X (arcsecs)

-300

-280

-260

-240

-220

-200

-180

Y (

arcs

ecs)

RHESSI 30-80 keVRHESSI 6-8 keV

AIA 193A

10 100energy [keV]

01

10

100

1000

10000

100000

phot

ons

s-1 c

m-2 k

eV-1

thermalabove-the-loop-top

footpoint

Figure 1 Summary of the RHESSI imaging spectroscopy observations of the 2012 July 19 flare On the left the time profile of the non-thermal HXR flux at 30ndash80 keVis given in blue with the linear GOES curve shown schematically in gray in the background The time interval used to reconstruct the RHESSI image shown in thecenter panel is marked in blue outlining the first HXR peak during the impulsive phase of the flare The thermal emissions in the 6ndash8 keV range (green contours at20 35 50 65 70 95) show the location of the main flare loops also seen in the 193 Aring AIA image The non-thermal HXR emissions come from thefootpoints of the thermal flare loops (blue contours at 30 50 70 90) but also from above the main flare loop as outlined by the 30ndash80 keV contours Relativeto the brightest foopoint the extended coronal source is shown in contours at 2 25 3 The plot on the right gives the imaging spectroscopy results The blackhistogram gives the imaging spectroscopy results for the combined coronal sources the observed footpoint spectrum is given by crosses The green and red curves arethe thermal and non-thermal fit to the combined coronal sources while the power-law fit to the footpoints is given in blue(A color version of this figure is available in the online journal)

the emission is intrinsically faint The drastically increasedcoverage and sensitivity provided by the Atmospheric ImagingAssembly (AIA Lemen et al 2012) on board the Solar DynamicObservatory (SDO) provides by far the best diagnostics ofthermal plasma In this paper we present observations of a GOESM9 class limb-flare on 2012 July 19 simultaneously observed byRHESSI and AIA While this remarkable event provides manyfascinating insights into flare physics (eg Liu et al 2013 Liu2013) our paper focuses only on the ambient density estimateswithin the acceleration region during the impulsive phase ofthe event For a detailed analysis of all phase of this flare werefer to Liu et al (2013) and Liu (2013) We first discuss theRHESSI imaging spectroscopy observations of the above-the-loop-top source during the main HXR peak followed by thederivation of the differential emission measure (DEM) fromAIA images and the density of the above-the-loop-top sourceIn Section 23 we use the derived ambient density to estimate thedensity of accelerated electrons within the acceleration regionto investigate the acceleration efficiency at the peak time of theHXR emission

2 OBSERVATIONS

The limb flare of SOL2012-07-19T0558 shows one of themost prominent above-the-loop-top HXR sources in the entireRHESSI data base (Figure 1) The coronal source is clearlyvisible in the 30ndash80 keV band for the whole duration of themain HXR peaks (0520-0527 UT) While the above-the-loop-top source is already visible in regular CLEAN images (Hurfordet al 2002) the images shown in Figure 1 are produced using thetwo-step CLEAN algorithm (Krucker et al 2011) The sourcelocation is sim35primeprime above the center of the thermal flare loopsseen by RHESSI in the 6ndash8 keV range and by the AIA 193 Aringwavelength channel one of the filters with high temperatureresponse Liu et al (2013) reported a smaller separation of21primeprime but this difference could be due to the different energyrange selection In either case the separation is even moreprominent than in the Masuda flare (Masuda et al 1994) The

RHESSI images also reveal the existence of footpoint sourcesThe northern footpoint dominates over the southern footpointbut this asymmetry is likely because part of the emission fromthe flare ribbons is occulted by the solar disk STEREO EUVIimages reveal that if seen from Earth the solar disk occultsa larger part of the southern ribbon than the northern ribbon(Patsourakos et al 2013 Liu et al 2013) indicating the weakersouthern footpoint could indeed be due to occultation Whilethe footpoints show the typical compact sources the above-the-loop-top sources is spatially extended From the CLEAN imageshown in Figure 1 we derived a deconvolved source diameter of15primeprime (for details in source size determination with RHESSI seeDennis amp Pernak 2009 and Warmuth amp Mann 2013) The limitedquality of the RHESSI reconstructed images does not allow usto derive fine structures within the above-the-loop-top sourceand the derived source size has to be understood as the secondmoment of the actual source shape Despite the low intensityof the coronal source its large size compared to the footpointareas makes its flux about a third (32 plusmn 3) of the total HXRflux This rather large fraction of non-thermal emission from thecorona could again be partially attributed to occultation of theflare ribbons as was also speculated for the Masuda flare (Wanget al 1995)

21 RHESSI Imaging Spectrocopy

Images below sim14 keV show the thermal flare loop only(Figure 2) and we therefore use the spatially integrated spectrumbelow 14 keV to derive the temperature of the main flare loops(T sim 23 MK and EM sim 4 times 1048 cmminus3 at 0521UT for thetemporal evolution see Liu et al 2013) Assuming a sphericalvolume (V sim 7 times 1026 cm3) the density of the thermal loopbecomes nth sim 8 times 1010 cmminus3 a typical value for flare loops(eg Caspi et al 2013) To get a separate spectrum of thechromospheric and coronal sources we use RHESSI standardimaging spectroscopy techniques (eg Krucker amp Lin 2002Battaglia amp Benz 2006) Above sim22 keV images show thefootpoints and the above-the-loop-top source but not the mainthermal loop The spectra derived from these images can be each

2

13T1236) respectively Overview time profiles and images aregiven in Figures 2 and 3 In the following section we describethe individual events and follow with a discussion of the resultsas summarized in Table 1

21 SOL2012-07-19T0558 (M77)

There are several previously published papers on this two-ribbon flare which appears as a textbook example of an arcadeat the western limb (Liu 2013 Liu et al 2013 Battaglia ampKontar 2013 Kirichek et al 2013 Patsourakos et al 2013

Krucker amp Battaglia 2014 Dudiacutek et al 2014 Sun et al 2014)Besides emission from the ribbons (Figure 3 left) the RHESSIobservations reveal a very prominent above-the-loop-top HXRsource that outlines the location of particle acceleration(Krucker amp Battaglia 2014) The STEREO images show thatthe flare ribbons are very close to the limb as seen from Earth(Figure 4 left) The southern ribbon shows significantlyweaker WL and HXR emissions than the northern one(Figure 3) This could be due to occultation as the southernribbon extends farther behind the limb and emission from thefar end of the ribbon could be hidden but the EUV emission

Figure 2 Time profiles of the three flares the top panels show the time profiles of the WL flux (summed over enhancement arbitrary units) and non-thermal HXRflux at 30ndash80 keV in red and blue respectively with the linear GOES curve shown schematically in gray in the background for timing reference Below the apparentaltitude of the peak of the WL emission above the limb is shown in red The black dotted lines give the FWHM of a Gaussian fit to the HMI source above the limb thethin black line is the deconvolved source size derived by a rough deconvolution of the observed size with the HMI PSF from Yeo et al (2014) The blue points givethe HXR centroid positions with error bars from statistical uncertainties No error bars are shown for the HMI centroids as they are dominated by systematicuncertainties (sim70 km)

Figure 3 X-ray and optical imaging of the three flares at the peak time of the impulsive phase the images are from HMI with the pre-flare image subtracted enhancedemission is shown in black The contours represent RHESSI Clean maps in the thermal (red 12ndash15 keV) and non-thermal (blue 30ndash80 keV) HXR range Two flares(left and right) are large-scale flares with widely separated ribbons and flare loops reaching high altitudes while the other flare (middle) is more compact For all flaresthe ribbons appear above the limb The large-scale flares show a non-thermal above-the-loop-top source

4

The Astrophysical Journal 80219 (10pp) 2015 March 20 Krucker et al

Selected scientific studies of this flare bull Patsourakos Vourlidas amp Stenborg 2013 ApJ 764

125 Prior confined eruption produced a pre-existing coronal flux rope which then erupted to give the M77 flare

bull Wei Liu Chen amp Petrosian 2013 ApJ 767 168 Detailed timeline of sequence of events including timing and propagation of EUV and X-ray sources

bull Rui Liu 2013 MNRAS 434 1309 Same onset time for HXR and microwave (Nobeyama RH data) bursts

bull Kruumlcker amp Battaglia 2014 ApJ 780 107 Ratio of thermal protons (from AIA) to non-thermal electrons (from RHESSI HXR) in loop-top source is of order 1

bull Sun Cheng amp Ding 2014 ApJ 786 73 Performed a very detailed DEM (imaging) analysis of this flare They used xrt_iterative_dem2pro (code by M Weber non-linear least square inversion with splines)

bull Kruumlcker et al 2015 ApJ 802 19 HMI white light (WL) enhancement at footprint co-incident with HXR footpoint source Interpretation energy deposition by non-thermal electrons is radiated away and does not cause chromospheric evaporation

M77 flare on 2012-07-19

Science Case 2) Reconnection Outflows

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

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easu

re [c

m-5

]

Emis

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sure

-wei

ghte

d lo

g T

Tota

l Em

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re [c

m-5

]

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-wei

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d lo

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m-5

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-wei

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d lo

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m-5

]

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Science Cases

bull Thermal evolution of active regions from emergence to decay

bull Thermodynamic structure of reconnection outflows

bull From chromospheric evaporation to catastrophic condensation

Lower right panel only Greyscale Blos from HMI Green 6MK EM YellowRed 10 MK EM

Other panels EM in various log T bins

DEM movie of the emergence of AR 11726

Science Case 1) Emerging Flux

Black pixels are where no solutions were found

DEM movie of the emergence of AR 11158

The Astrophysical Journal 780107 (6pp) 2014 January 1 Krucker amp Battaglia

0515 0520 0525 0530 0535UT on 2012-Jul-19

0

20

40

60

80

100

120

coun

ts s

-1

RHESSI 30-80 keV

GOESgt3 keV

900 920 940 960 980 1000 1020X (arcsecs)

-300

-280

-260

-240

-220

-200

-180

Y (

arcs

ecs)

RHESSI 30-80 keVRHESSI 6-8 keV

AIA 193A

10 100energy [keV]

01

10

100

1000

10000

100000

phot

ons

s-1 c

m-2 k

eV-1

thermalabove-the-loop-top

footpoint

Figure 1 Summary of the RHESSI imaging spectroscopy observations of the 2012 July 19 flare On the left the time profile of the non-thermal HXR flux at 30ndash80 keVis given in blue with the linear GOES curve shown schematically in gray in the background The time interval used to reconstruct the RHESSI image shown in thecenter panel is marked in blue outlining the first HXR peak during the impulsive phase of the flare The thermal emissions in the 6ndash8 keV range (green contours at20 35 50 65 70 95) show the location of the main flare loops also seen in the 193 Aring AIA image The non-thermal HXR emissions come from thefootpoints of the thermal flare loops (blue contours at 30 50 70 90) but also from above the main flare loop as outlined by the 30ndash80 keV contours Relativeto the brightest foopoint the extended coronal source is shown in contours at 2 25 3 The plot on the right gives the imaging spectroscopy results The blackhistogram gives the imaging spectroscopy results for the combined coronal sources the observed footpoint spectrum is given by crosses The green and red curves arethe thermal and non-thermal fit to the combined coronal sources while the power-law fit to the footpoints is given in blue(A color version of this figure is available in the online journal)

the emission is intrinsically faint The drastically increasedcoverage and sensitivity provided by the Atmospheric ImagingAssembly (AIA Lemen et al 2012) on board the Solar DynamicObservatory (SDO) provides by far the best diagnostics ofthermal plasma In this paper we present observations of a GOESM9 class limb-flare on 2012 July 19 simultaneously observed byRHESSI and AIA While this remarkable event provides manyfascinating insights into flare physics (eg Liu et al 2013 Liu2013) our paper focuses only on the ambient density estimateswithin the acceleration region during the impulsive phase ofthe event For a detailed analysis of all phase of this flare werefer to Liu et al (2013) and Liu (2013) We first discuss theRHESSI imaging spectroscopy observations of the above-the-loop-top source during the main HXR peak followed by thederivation of the differential emission measure (DEM) fromAIA images and the density of the above-the-loop-top sourceIn Section 23 we use the derived ambient density to estimate thedensity of accelerated electrons within the acceleration regionto investigate the acceleration efficiency at the peak time of theHXR emission

2 OBSERVATIONS

The limb flare of SOL2012-07-19T0558 shows one of themost prominent above-the-loop-top HXR sources in the entireRHESSI data base (Figure 1) The coronal source is clearlyvisible in the 30ndash80 keV band for the whole duration of themain HXR peaks (0520-0527 UT) While the above-the-loop-top source is already visible in regular CLEAN images (Hurfordet al 2002) the images shown in Figure 1 are produced using thetwo-step CLEAN algorithm (Krucker et al 2011) The sourcelocation is sim35primeprime above the center of the thermal flare loopsseen by RHESSI in the 6ndash8 keV range and by the AIA 193 Aringwavelength channel one of the filters with high temperatureresponse Liu et al (2013) reported a smaller separation of21primeprime but this difference could be due to the different energyrange selection In either case the separation is even moreprominent than in the Masuda flare (Masuda et al 1994) The

RHESSI images also reveal the existence of footpoint sourcesThe northern footpoint dominates over the southern footpointbut this asymmetry is likely because part of the emission fromthe flare ribbons is occulted by the solar disk STEREO EUVIimages reveal that if seen from Earth the solar disk occultsa larger part of the southern ribbon than the northern ribbon(Patsourakos et al 2013 Liu et al 2013) indicating the weakersouthern footpoint could indeed be due to occultation Whilethe footpoints show the typical compact sources the above-the-loop-top sources is spatially extended From the CLEAN imageshown in Figure 1 we derived a deconvolved source diameter of15primeprime (for details in source size determination with RHESSI seeDennis amp Pernak 2009 and Warmuth amp Mann 2013) The limitedquality of the RHESSI reconstructed images does not allow usto derive fine structures within the above-the-loop-top sourceand the derived source size has to be understood as the secondmoment of the actual source shape Despite the low intensityof the coronal source its large size compared to the footpointareas makes its flux about a third (32 plusmn 3) of the total HXRflux This rather large fraction of non-thermal emission from thecorona could again be partially attributed to occultation of theflare ribbons as was also speculated for the Masuda flare (Wanget al 1995)

21 RHESSI Imaging Spectrocopy

Images below sim14 keV show the thermal flare loop only(Figure 2) and we therefore use the spatially integrated spectrumbelow 14 keV to derive the temperature of the main flare loops(T sim 23 MK and EM sim 4 times 1048 cmminus3 at 0521UT for thetemporal evolution see Liu et al 2013) Assuming a sphericalvolume (V sim 7 times 1026 cm3) the density of the thermal loopbecomes nth sim 8 times 1010 cmminus3 a typical value for flare loops(eg Caspi et al 2013) To get a separate spectrum of thechromospheric and coronal sources we use RHESSI standardimaging spectroscopy techniques (eg Krucker amp Lin 2002Battaglia amp Benz 2006) Above sim22 keV images show thefootpoints and the above-the-loop-top source but not the mainthermal loop The spectra derived from these images can be each

2

13T1236) respectively Overview time profiles and images aregiven in Figures 2 and 3 In the following section we describethe individual events and follow with a discussion of the resultsas summarized in Table 1

21 SOL2012-07-19T0558 (M77)

There are several previously published papers on this two-ribbon flare which appears as a textbook example of an arcadeat the western limb (Liu 2013 Liu et al 2013 Battaglia ampKontar 2013 Kirichek et al 2013 Patsourakos et al 2013

Krucker amp Battaglia 2014 Dudiacutek et al 2014 Sun et al 2014)Besides emission from the ribbons (Figure 3 left) the RHESSIobservations reveal a very prominent above-the-loop-top HXRsource that outlines the location of particle acceleration(Krucker amp Battaglia 2014) The STEREO images show thatthe flare ribbons are very close to the limb as seen from Earth(Figure 4 left) The southern ribbon shows significantlyweaker WL and HXR emissions than the northern one(Figure 3) This could be due to occultation as the southernribbon extends farther behind the limb and emission from thefar end of the ribbon could be hidden but the EUV emission

Figure 2 Time profiles of the three flares the top panels show the time profiles of the WL flux (summed over enhancement arbitrary units) and non-thermal HXRflux at 30ndash80 keV in red and blue respectively with the linear GOES curve shown schematically in gray in the background for timing reference Below the apparentaltitude of the peak of the WL emission above the limb is shown in red The black dotted lines give the FWHM of a Gaussian fit to the HMI source above the limb thethin black line is the deconvolved source size derived by a rough deconvolution of the observed size with the HMI PSF from Yeo et al (2014) The blue points givethe HXR centroid positions with error bars from statistical uncertainties No error bars are shown for the HMI centroids as they are dominated by systematicuncertainties (sim70 km)

Figure 3 X-ray and optical imaging of the three flares at the peak time of the impulsive phase the images are from HMI with the pre-flare image subtracted enhancedemission is shown in black The contours represent RHESSI Clean maps in the thermal (red 12ndash15 keV) and non-thermal (blue 30ndash80 keV) HXR range Two flares(left and right) are large-scale flares with widely separated ribbons and flare loops reaching high altitudes while the other flare (middle) is more compact For all flaresthe ribbons appear above the limb The large-scale flares show a non-thermal above-the-loop-top source

4

The Astrophysical Journal 80219 (10pp) 2015 March 20 Krucker et al

Selected scientific studies of this flare bull Patsourakos Vourlidas amp Stenborg 2013 ApJ 764

125 Prior confined eruption produced a pre-existing coronal flux rope which then erupted to give the M77 flare

bull Wei Liu Chen amp Petrosian 2013 ApJ 767 168 Detailed timeline of sequence of events including timing and propagation of EUV and X-ray sources

bull Rui Liu 2013 MNRAS 434 1309 Same onset time for HXR and microwave (Nobeyama RH data) bursts

bull Kruumlcker amp Battaglia 2014 ApJ 780 107 Ratio of thermal protons (from AIA) to non-thermal electrons (from RHESSI HXR) in loop-top source is of order 1

bull Sun Cheng amp Ding 2014 ApJ 786 73 Performed a very detailed DEM (imaging) analysis of this flare They used xrt_iterative_dem2pro (code by M Weber non-linear least square inversion with splines)

bull Kruumlcker et al 2015 ApJ 802 19 HMI white light (WL) enhancement at footprint co-incident with HXR footpoint source Interpretation energy deposition by non-thermal electrons is radiated away and does not cause chromospheric evaporation

M77 flare on 2012-07-19

Science Case 2) Reconnection Outflows

Emis

sion

Mea

sure

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ghte

d lo

g T

Tota

l Em

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-wei

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d lo

g T

Tota

l Em

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]

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sure

-wei

ghte

d lo

g T

Tota

l Em

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re [c

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Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Lower right panel only Greyscale Blos from HMI Green 6MK EM YellowRed 10 MK EM

Other panels EM in various log T bins

DEM movie of the emergence of AR 11726

Science Case 1) Emerging Flux

Black pixels are where no solutions were found

DEM movie of the emergence of AR 11158

The Astrophysical Journal 780107 (6pp) 2014 January 1 Krucker amp Battaglia

0515 0520 0525 0530 0535UT on 2012-Jul-19

0

20

40

60

80

100

120

coun

ts s

-1

RHESSI 30-80 keV

GOESgt3 keV

900 920 940 960 980 1000 1020X (arcsecs)

-300

-280

-260

-240

-220

-200

-180

Y (

arcs

ecs)

RHESSI 30-80 keVRHESSI 6-8 keV

AIA 193A

10 100energy [keV]

01

10

100

1000

10000

100000

phot

ons

s-1 c

m-2 k

eV-1

thermalabove-the-loop-top

footpoint

Figure 1 Summary of the RHESSI imaging spectroscopy observations of the 2012 July 19 flare On the left the time profile of the non-thermal HXR flux at 30ndash80 keVis given in blue with the linear GOES curve shown schematically in gray in the background The time interval used to reconstruct the RHESSI image shown in thecenter panel is marked in blue outlining the first HXR peak during the impulsive phase of the flare The thermal emissions in the 6ndash8 keV range (green contours at20 35 50 65 70 95) show the location of the main flare loops also seen in the 193 Aring AIA image The non-thermal HXR emissions come from thefootpoints of the thermal flare loops (blue contours at 30 50 70 90) but also from above the main flare loop as outlined by the 30ndash80 keV contours Relativeto the brightest foopoint the extended coronal source is shown in contours at 2 25 3 The plot on the right gives the imaging spectroscopy results The blackhistogram gives the imaging spectroscopy results for the combined coronal sources the observed footpoint spectrum is given by crosses The green and red curves arethe thermal and non-thermal fit to the combined coronal sources while the power-law fit to the footpoints is given in blue(A color version of this figure is available in the online journal)

the emission is intrinsically faint The drastically increasedcoverage and sensitivity provided by the Atmospheric ImagingAssembly (AIA Lemen et al 2012) on board the Solar DynamicObservatory (SDO) provides by far the best diagnostics ofthermal plasma In this paper we present observations of a GOESM9 class limb-flare on 2012 July 19 simultaneously observed byRHESSI and AIA While this remarkable event provides manyfascinating insights into flare physics (eg Liu et al 2013 Liu2013) our paper focuses only on the ambient density estimateswithin the acceleration region during the impulsive phase ofthe event For a detailed analysis of all phase of this flare werefer to Liu et al (2013) and Liu (2013) We first discuss theRHESSI imaging spectroscopy observations of the above-the-loop-top source during the main HXR peak followed by thederivation of the differential emission measure (DEM) fromAIA images and the density of the above-the-loop-top sourceIn Section 23 we use the derived ambient density to estimate thedensity of accelerated electrons within the acceleration regionto investigate the acceleration efficiency at the peak time of theHXR emission

2 OBSERVATIONS

The limb flare of SOL2012-07-19T0558 shows one of themost prominent above-the-loop-top HXR sources in the entireRHESSI data base (Figure 1) The coronal source is clearlyvisible in the 30ndash80 keV band for the whole duration of themain HXR peaks (0520-0527 UT) While the above-the-loop-top source is already visible in regular CLEAN images (Hurfordet al 2002) the images shown in Figure 1 are produced using thetwo-step CLEAN algorithm (Krucker et al 2011) The sourcelocation is sim35primeprime above the center of the thermal flare loopsseen by RHESSI in the 6ndash8 keV range and by the AIA 193 Aringwavelength channel one of the filters with high temperatureresponse Liu et al (2013) reported a smaller separation of21primeprime but this difference could be due to the different energyrange selection In either case the separation is even moreprominent than in the Masuda flare (Masuda et al 1994) The

RHESSI images also reveal the existence of footpoint sourcesThe northern footpoint dominates over the southern footpointbut this asymmetry is likely because part of the emission fromthe flare ribbons is occulted by the solar disk STEREO EUVIimages reveal that if seen from Earth the solar disk occultsa larger part of the southern ribbon than the northern ribbon(Patsourakos et al 2013 Liu et al 2013) indicating the weakersouthern footpoint could indeed be due to occultation Whilethe footpoints show the typical compact sources the above-the-loop-top sources is spatially extended From the CLEAN imageshown in Figure 1 we derived a deconvolved source diameter of15primeprime (for details in source size determination with RHESSI seeDennis amp Pernak 2009 and Warmuth amp Mann 2013) The limitedquality of the RHESSI reconstructed images does not allow usto derive fine structures within the above-the-loop-top sourceand the derived source size has to be understood as the secondmoment of the actual source shape Despite the low intensityof the coronal source its large size compared to the footpointareas makes its flux about a third (32 plusmn 3) of the total HXRflux This rather large fraction of non-thermal emission from thecorona could again be partially attributed to occultation of theflare ribbons as was also speculated for the Masuda flare (Wanget al 1995)

21 RHESSI Imaging Spectrocopy

Images below sim14 keV show the thermal flare loop only(Figure 2) and we therefore use the spatially integrated spectrumbelow 14 keV to derive the temperature of the main flare loops(T sim 23 MK and EM sim 4 times 1048 cmminus3 at 0521UT for thetemporal evolution see Liu et al 2013) Assuming a sphericalvolume (V sim 7 times 1026 cm3) the density of the thermal loopbecomes nth sim 8 times 1010 cmminus3 a typical value for flare loops(eg Caspi et al 2013) To get a separate spectrum of thechromospheric and coronal sources we use RHESSI standardimaging spectroscopy techniques (eg Krucker amp Lin 2002Battaglia amp Benz 2006) Above sim22 keV images show thefootpoints and the above-the-loop-top source but not the mainthermal loop The spectra derived from these images can be each

2

13T1236) respectively Overview time profiles and images aregiven in Figures 2 and 3 In the following section we describethe individual events and follow with a discussion of the resultsas summarized in Table 1

21 SOL2012-07-19T0558 (M77)

There are several previously published papers on this two-ribbon flare which appears as a textbook example of an arcadeat the western limb (Liu 2013 Liu et al 2013 Battaglia ampKontar 2013 Kirichek et al 2013 Patsourakos et al 2013

Krucker amp Battaglia 2014 Dudiacutek et al 2014 Sun et al 2014)Besides emission from the ribbons (Figure 3 left) the RHESSIobservations reveal a very prominent above-the-loop-top HXRsource that outlines the location of particle acceleration(Krucker amp Battaglia 2014) The STEREO images show thatthe flare ribbons are very close to the limb as seen from Earth(Figure 4 left) The southern ribbon shows significantlyweaker WL and HXR emissions than the northern one(Figure 3) This could be due to occultation as the southernribbon extends farther behind the limb and emission from thefar end of the ribbon could be hidden but the EUV emission

Figure 2 Time profiles of the three flares the top panels show the time profiles of the WL flux (summed over enhancement arbitrary units) and non-thermal HXRflux at 30ndash80 keV in red and blue respectively with the linear GOES curve shown schematically in gray in the background for timing reference Below the apparentaltitude of the peak of the WL emission above the limb is shown in red The black dotted lines give the FWHM of a Gaussian fit to the HMI source above the limb thethin black line is the deconvolved source size derived by a rough deconvolution of the observed size with the HMI PSF from Yeo et al (2014) The blue points givethe HXR centroid positions with error bars from statistical uncertainties No error bars are shown for the HMI centroids as they are dominated by systematicuncertainties (sim70 km)

Figure 3 X-ray and optical imaging of the three flares at the peak time of the impulsive phase the images are from HMI with the pre-flare image subtracted enhancedemission is shown in black The contours represent RHESSI Clean maps in the thermal (red 12ndash15 keV) and non-thermal (blue 30ndash80 keV) HXR range Two flares(left and right) are large-scale flares with widely separated ribbons and flare loops reaching high altitudes while the other flare (middle) is more compact For all flaresthe ribbons appear above the limb The large-scale flares show a non-thermal above-the-loop-top source

4

The Astrophysical Journal 80219 (10pp) 2015 March 20 Krucker et al

Selected scientific studies of this flare bull Patsourakos Vourlidas amp Stenborg 2013 ApJ 764

125 Prior confined eruption produced a pre-existing coronal flux rope which then erupted to give the M77 flare

bull Wei Liu Chen amp Petrosian 2013 ApJ 767 168 Detailed timeline of sequence of events including timing and propagation of EUV and X-ray sources

bull Rui Liu 2013 MNRAS 434 1309 Same onset time for HXR and microwave (Nobeyama RH data) bursts

bull Kruumlcker amp Battaglia 2014 ApJ 780 107 Ratio of thermal protons (from AIA) to non-thermal electrons (from RHESSI HXR) in loop-top source is of order 1

bull Sun Cheng amp Ding 2014 ApJ 786 73 Performed a very detailed DEM (imaging) analysis of this flare They used xrt_iterative_dem2pro (code by M Weber non-linear least square inversion with splines)

bull Kruumlcker et al 2015 ApJ 802 19 HMI white light (WL) enhancement at footprint co-incident with HXR footpoint source Interpretation energy deposition by non-thermal electrons is radiated away and does not cause chromospheric evaporation

M77 flare on 2012-07-19

Science Case 2) Reconnection Outflows

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Black pixels are where no solutions were found

DEM movie of the emergence of AR 11158

The Astrophysical Journal 780107 (6pp) 2014 January 1 Krucker amp Battaglia

0515 0520 0525 0530 0535UT on 2012-Jul-19

0

20

40

60

80

100

120

coun

ts s

-1

RHESSI 30-80 keV

GOESgt3 keV

900 920 940 960 980 1000 1020X (arcsecs)

-300

-280

-260

-240

-220

-200

-180

Y (

arcs

ecs)

RHESSI 30-80 keVRHESSI 6-8 keV

AIA 193A

10 100energy [keV]

01

10

100

1000

10000

100000

phot

ons

s-1 c

m-2 k

eV-1

thermalabove-the-loop-top

footpoint

Figure 1 Summary of the RHESSI imaging spectroscopy observations of the 2012 July 19 flare On the left the time profile of the non-thermal HXR flux at 30ndash80 keVis given in blue with the linear GOES curve shown schematically in gray in the background The time interval used to reconstruct the RHESSI image shown in thecenter panel is marked in blue outlining the first HXR peak during the impulsive phase of the flare The thermal emissions in the 6ndash8 keV range (green contours at20 35 50 65 70 95) show the location of the main flare loops also seen in the 193 Aring AIA image The non-thermal HXR emissions come from thefootpoints of the thermal flare loops (blue contours at 30 50 70 90) but also from above the main flare loop as outlined by the 30ndash80 keV contours Relativeto the brightest foopoint the extended coronal source is shown in contours at 2 25 3 The plot on the right gives the imaging spectroscopy results The blackhistogram gives the imaging spectroscopy results for the combined coronal sources the observed footpoint spectrum is given by crosses The green and red curves arethe thermal and non-thermal fit to the combined coronal sources while the power-law fit to the footpoints is given in blue(A color version of this figure is available in the online journal)

the emission is intrinsically faint The drastically increasedcoverage and sensitivity provided by the Atmospheric ImagingAssembly (AIA Lemen et al 2012) on board the Solar DynamicObservatory (SDO) provides by far the best diagnostics ofthermal plasma In this paper we present observations of a GOESM9 class limb-flare on 2012 July 19 simultaneously observed byRHESSI and AIA While this remarkable event provides manyfascinating insights into flare physics (eg Liu et al 2013 Liu2013) our paper focuses only on the ambient density estimateswithin the acceleration region during the impulsive phase ofthe event For a detailed analysis of all phase of this flare werefer to Liu et al (2013) and Liu (2013) We first discuss theRHESSI imaging spectroscopy observations of the above-the-loop-top source during the main HXR peak followed by thederivation of the differential emission measure (DEM) fromAIA images and the density of the above-the-loop-top sourceIn Section 23 we use the derived ambient density to estimate thedensity of accelerated electrons within the acceleration regionto investigate the acceleration efficiency at the peak time of theHXR emission

2 OBSERVATIONS

The limb flare of SOL2012-07-19T0558 shows one of themost prominent above-the-loop-top HXR sources in the entireRHESSI data base (Figure 1) The coronal source is clearlyvisible in the 30ndash80 keV band for the whole duration of themain HXR peaks (0520-0527 UT) While the above-the-loop-top source is already visible in regular CLEAN images (Hurfordet al 2002) the images shown in Figure 1 are produced using thetwo-step CLEAN algorithm (Krucker et al 2011) The sourcelocation is sim35primeprime above the center of the thermal flare loopsseen by RHESSI in the 6ndash8 keV range and by the AIA 193 Aringwavelength channel one of the filters with high temperatureresponse Liu et al (2013) reported a smaller separation of21primeprime but this difference could be due to the different energyrange selection In either case the separation is even moreprominent than in the Masuda flare (Masuda et al 1994) The

RHESSI images also reveal the existence of footpoint sourcesThe northern footpoint dominates over the southern footpointbut this asymmetry is likely because part of the emission fromthe flare ribbons is occulted by the solar disk STEREO EUVIimages reveal that if seen from Earth the solar disk occultsa larger part of the southern ribbon than the northern ribbon(Patsourakos et al 2013 Liu et al 2013) indicating the weakersouthern footpoint could indeed be due to occultation Whilethe footpoints show the typical compact sources the above-the-loop-top sources is spatially extended From the CLEAN imageshown in Figure 1 we derived a deconvolved source diameter of15primeprime (for details in source size determination with RHESSI seeDennis amp Pernak 2009 and Warmuth amp Mann 2013) The limitedquality of the RHESSI reconstructed images does not allow usto derive fine structures within the above-the-loop-top sourceand the derived source size has to be understood as the secondmoment of the actual source shape Despite the low intensityof the coronal source its large size compared to the footpointareas makes its flux about a third (32 plusmn 3) of the total HXRflux This rather large fraction of non-thermal emission from thecorona could again be partially attributed to occultation of theflare ribbons as was also speculated for the Masuda flare (Wanget al 1995)

21 RHESSI Imaging Spectrocopy

Images below sim14 keV show the thermal flare loop only(Figure 2) and we therefore use the spatially integrated spectrumbelow 14 keV to derive the temperature of the main flare loops(T sim 23 MK and EM sim 4 times 1048 cmminus3 at 0521UT for thetemporal evolution see Liu et al 2013) Assuming a sphericalvolume (V sim 7 times 1026 cm3) the density of the thermal loopbecomes nth sim 8 times 1010 cmminus3 a typical value for flare loops(eg Caspi et al 2013) To get a separate spectrum of thechromospheric and coronal sources we use RHESSI standardimaging spectroscopy techniques (eg Krucker amp Lin 2002Battaglia amp Benz 2006) Above sim22 keV images show thefootpoints and the above-the-loop-top source but not the mainthermal loop The spectra derived from these images can be each

2

13T1236) respectively Overview time profiles and images aregiven in Figures 2 and 3 In the following section we describethe individual events and follow with a discussion of the resultsas summarized in Table 1

21 SOL2012-07-19T0558 (M77)

There are several previously published papers on this two-ribbon flare which appears as a textbook example of an arcadeat the western limb (Liu 2013 Liu et al 2013 Battaglia ampKontar 2013 Kirichek et al 2013 Patsourakos et al 2013

Krucker amp Battaglia 2014 Dudiacutek et al 2014 Sun et al 2014)Besides emission from the ribbons (Figure 3 left) the RHESSIobservations reveal a very prominent above-the-loop-top HXRsource that outlines the location of particle acceleration(Krucker amp Battaglia 2014) The STEREO images show thatthe flare ribbons are very close to the limb as seen from Earth(Figure 4 left) The southern ribbon shows significantlyweaker WL and HXR emissions than the northern one(Figure 3) This could be due to occultation as the southernribbon extends farther behind the limb and emission from thefar end of the ribbon could be hidden but the EUV emission

Figure 2 Time profiles of the three flares the top panels show the time profiles of the WL flux (summed over enhancement arbitrary units) and non-thermal HXRflux at 30ndash80 keV in red and blue respectively with the linear GOES curve shown schematically in gray in the background for timing reference Below the apparentaltitude of the peak of the WL emission above the limb is shown in red The black dotted lines give the FWHM of a Gaussian fit to the HMI source above the limb thethin black line is the deconvolved source size derived by a rough deconvolution of the observed size with the HMI PSF from Yeo et al (2014) The blue points givethe HXR centroid positions with error bars from statistical uncertainties No error bars are shown for the HMI centroids as they are dominated by systematicuncertainties (sim70 km)

Figure 3 X-ray and optical imaging of the three flares at the peak time of the impulsive phase the images are from HMI with the pre-flare image subtracted enhancedemission is shown in black The contours represent RHESSI Clean maps in the thermal (red 12ndash15 keV) and non-thermal (blue 30ndash80 keV) HXR range Two flares(left and right) are large-scale flares with widely separated ribbons and flare loops reaching high altitudes while the other flare (middle) is more compact For all flaresthe ribbons appear above the limb The large-scale flares show a non-thermal above-the-loop-top source

4

The Astrophysical Journal 80219 (10pp) 2015 March 20 Krucker et al

Selected scientific studies of this flare bull Patsourakos Vourlidas amp Stenborg 2013 ApJ 764

125 Prior confined eruption produced a pre-existing coronal flux rope which then erupted to give the M77 flare

bull Wei Liu Chen amp Petrosian 2013 ApJ 767 168 Detailed timeline of sequence of events including timing and propagation of EUV and X-ray sources

bull Rui Liu 2013 MNRAS 434 1309 Same onset time for HXR and microwave (Nobeyama RH data) bursts

bull Kruumlcker amp Battaglia 2014 ApJ 780 107 Ratio of thermal protons (from AIA) to non-thermal electrons (from RHESSI HXR) in loop-top source is of order 1

bull Sun Cheng amp Ding 2014 ApJ 786 73 Performed a very detailed DEM (imaging) analysis of this flare They used xrt_iterative_dem2pro (code by M Weber non-linear least square inversion with splines)

bull Kruumlcker et al 2015 ApJ 802 19 HMI white light (WL) enhancement at footprint co-incident with HXR footpoint source Interpretation energy deposition by non-thermal electrons is radiated away and does not cause chromospheric evaporation

M77 flare on 2012-07-19

Science Case 2) Reconnection Outflows

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

The Astrophysical Journal 780107 (6pp) 2014 January 1 Krucker amp Battaglia

0515 0520 0525 0530 0535UT on 2012-Jul-19

0

20

40

60

80

100

120

coun

ts s

-1

RHESSI 30-80 keV

GOESgt3 keV

900 920 940 960 980 1000 1020X (arcsecs)

-300

-280

-260

-240

-220

-200

-180

Y (

arcs

ecs)

RHESSI 30-80 keVRHESSI 6-8 keV

AIA 193A

10 100energy [keV]

01

10

100

1000

10000

100000

phot

ons

s-1 c

m-2 k

eV-1

thermalabove-the-loop-top

footpoint

Figure 1 Summary of the RHESSI imaging spectroscopy observations of the 2012 July 19 flare On the left the time profile of the non-thermal HXR flux at 30ndash80 keVis given in blue with the linear GOES curve shown schematically in gray in the background The time interval used to reconstruct the RHESSI image shown in thecenter panel is marked in blue outlining the first HXR peak during the impulsive phase of the flare The thermal emissions in the 6ndash8 keV range (green contours at20 35 50 65 70 95) show the location of the main flare loops also seen in the 193 Aring AIA image The non-thermal HXR emissions come from thefootpoints of the thermal flare loops (blue contours at 30 50 70 90) but also from above the main flare loop as outlined by the 30ndash80 keV contours Relativeto the brightest foopoint the extended coronal source is shown in contours at 2 25 3 The plot on the right gives the imaging spectroscopy results The blackhistogram gives the imaging spectroscopy results for the combined coronal sources the observed footpoint spectrum is given by crosses The green and red curves arethe thermal and non-thermal fit to the combined coronal sources while the power-law fit to the footpoints is given in blue(A color version of this figure is available in the online journal)

the emission is intrinsically faint The drastically increasedcoverage and sensitivity provided by the Atmospheric ImagingAssembly (AIA Lemen et al 2012) on board the Solar DynamicObservatory (SDO) provides by far the best diagnostics ofthermal plasma In this paper we present observations of a GOESM9 class limb-flare on 2012 July 19 simultaneously observed byRHESSI and AIA While this remarkable event provides manyfascinating insights into flare physics (eg Liu et al 2013 Liu2013) our paper focuses only on the ambient density estimateswithin the acceleration region during the impulsive phase ofthe event For a detailed analysis of all phase of this flare werefer to Liu et al (2013) and Liu (2013) We first discuss theRHESSI imaging spectroscopy observations of the above-the-loop-top source during the main HXR peak followed by thederivation of the differential emission measure (DEM) fromAIA images and the density of the above-the-loop-top sourceIn Section 23 we use the derived ambient density to estimate thedensity of accelerated electrons within the acceleration regionto investigate the acceleration efficiency at the peak time of theHXR emission

2 OBSERVATIONS

The limb flare of SOL2012-07-19T0558 shows one of themost prominent above-the-loop-top HXR sources in the entireRHESSI data base (Figure 1) The coronal source is clearlyvisible in the 30ndash80 keV band for the whole duration of themain HXR peaks (0520-0527 UT) While the above-the-loop-top source is already visible in regular CLEAN images (Hurfordet al 2002) the images shown in Figure 1 are produced using thetwo-step CLEAN algorithm (Krucker et al 2011) The sourcelocation is sim35primeprime above the center of the thermal flare loopsseen by RHESSI in the 6ndash8 keV range and by the AIA 193 Aringwavelength channel one of the filters with high temperatureresponse Liu et al (2013) reported a smaller separation of21primeprime but this difference could be due to the different energyrange selection In either case the separation is even moreprominent than in the Masuda flare (Masuda et al 1994) The

RHESSI images also reveal the existence of footpoint sourcesThe northern footpoint dominates over the southern footpointbut this asymmetry is likely because part of the emission fromthe flare ribbons is occulted by the solar disk STEREO EUVIimages reveal that if seen from Earth the solar disk occultsa larger part of the southern ribbon than the northern ribbon(Patsourakos et al 2013 Liu et al 2013) indicating the weakersouthern footpoint could indeed be due to occultation Whilethe footpoints show the typical compact sources the above-the-loop-top sources is spatially extended From the CLEAN imageshown in Figure 1 we derived a deconvolved source diameter of15primeprime (for details in source size determination with RHESSI seeDennis amp Pernak 2009 and Warmuth amp Mann 2013) The limitedquality of the RHESSI reconstructed images does not allow usto derive fine structures within the above-the-loop-top sourceand the derived source size has to be understood as the secondmoment of the actual source shape Despite the low intensityof the coronal source its large size compared to the footpointareas makes its flux about a third (32 plusmn 3) of the total HXRflux This rather large fraction of non-thermal emission from thecorona could again be partially attributed to occultation of theflare ribbons as was also speculated for the Masuda flare (Wanget al 1995)

21 RHESSI Imaging Spectrocopy

Images below sim14 keV show the thermal flare loop only(Figure 2) and we therefore use the spatially integrated spectrumbelow 14 keV to derive the temperature of the main flare loops(T sim 23 MK and EM sim 4 times 1048 cmminus3 at 0521UT for thetemporal evolution see Liu et al 2013) Assuming a sphericalvolume (V sim 7 times 1026 cm3) the density of the thermal loopbecomes nth sim 8 times 1010 cmminus3 a typical value for flare loops(eg Caspi et al 2013) To get a separate spectrum of thechromospheric and coronal sources we use RHESSI standardimaging spectroscopy techniques (eg Krucker amp Lin 2002Battaglia amp Benz 2006) Above sim22 keV images show thefootpoints and the above-the-loop-top source but not the mainthermal loop The spectra derived from these images can be each

2

13T1236) respectively Overview time profiles and images aregiven in Figures 2 and 3 In the following section we describethe individual events and follow with a discussion of the resultsas summarized in Table 1

21 SOL2012-07-19T0558 (M77)

There are several previously published papers on this two-ribbon flare which appears as a textbook example of an arcadeat the western limb (Liu 2013 Liu et al 2013 Battaglia ampKontar 2013 Kirichek et al 2013 Patsourakos et al 2013

Krucker amp Battaglia 2014 Dudiacutek et al 2014 Sun et al 2014)Besides emission from the ribbons (Figure 3 left) the RHESSIobservations reveal a very prominent above-the-loop-top HXRsource that outlines the location of particle acceleration(Krucker amp Battaglia 2014) The STEREO images show thatthe flare ribbons are very close to the limb as seen from Earth(Figure 4 left) The southern ribbon shows significantlyweaker WL and HXR emissions than the northern one(Figure 3) This could be due to occultation as the southernribbon extends farther behind the limb and emission from thefar end of the ribbon could be hidden but the EUV emission

Figure 2 Time profiles of the three flares the top panels show the time profiles of the WL flux (summed over enhancement arbitrary units) and non-thermal HXRflux at 30ndash80 keV in red and blue respectively with the linear GOES curve shown schematically in gray in the background for timing reference Below the apparentaltitude of the peak of the WL emission above the limb is shown in red The black dotted lines give the FWHM of a Gaussian fit to the HMI source above the limb thethin black line is the deconvolved source size derived by a rough deconvolution of the observed size with the HMI PSF from Yeo et al (2014) The blue points givethe HXR centroid positions with error bars from statistical uncertainties No error bars are shown for the HMI centroids as they are dominated by systematicuncertainties (sim70 km)

Figure 3 X-ray and optical imaging of the three flares at the peak time of the impulsive phase the images are from HMI with the pre-flare image subtracted enhancedemission is shown in black The contours represent RHESSI Clean maps in the thermal (red 12ndash15 keV) and non-thermal (blue 30ndash80 keV) HXR range Two flares(left and right) are large-scale flares with widely separated ribbons and flare loops reaching high altitudes while the other flare (middle) is more compact For all flaresthe ribbons appear above the limb The large-scale flares show a non-thermal above-the-loop-top source

4

The Astrophysical Journal 80219 (10pp) 2015 March 20 Krucker et al

Selected scientific studies of this flare bull Patsourakos Vourlidas amp Stenborg 2013 ApJ 764

125 Prior confined eruption produced a pre-existing coronal flux rope which then erupted to give the M77 flare

bull Wei Liu Chen amp Petrosian 2013 ApJ 767 168 Detailed timeline of sequence of events including timing and propagation of EUV and X-ray sources

bull Rui Liu 2013 MNRAS 434 1309 Same onset time for HXR and microwave (Nobeyama RH data) bursts

bull Kruumlcker amp Battaglia 2014 ApJ 780 107 Ratio of thermal protons (from AIA) to non-thermal electrons (from RHESSI HXR) in loop-top source is of order 1

bull Sun Cheng amp Ding 2014 ApJ 786 73 Performed a very detailed DEM (imaging) analysis of this flare They used xrt_iterative_dem2pro (code by M Weber non-linear least square inversion with splines)

bull Kruumlcker et al 2015 ApJ 802 19 HMI white light (WL) enhancement at footprint co-incident with HXR footpoint source Interpretation energy deposition by non-thermal electrons is radiated away and does not cause chromospheric evaporation

M77 flare on 2012-07-19

Science Case 2) Reconnection Outflows

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Emis

sion

Mea

sure

-wei

ghte

d lo

g T

Tota

l Em

issi

on M

easu

re [c

m-5

]

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation

of of the eruption of a pre-existing flux rope t r i g g e r e d b y fl u x emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Shiota et al (2005 ApJ 634 663) bull 25D MHD simulation of

of the eruption of a pre-existing flux rope triggered by flux emergence

bull Similar scenario as modeled by Chen amp Shibata (2000 ApJ 545 524) but with field-aligned thermal conduction

bull Both temperature and density are initially uniform (dimensionless value of unity)

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Dashed contours Total EM =1029 cm-5

Solid contours Total EM =1030 cm-5

Chromospheric evaporation

Downward mass pumping fromreconnection outflow

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

Influence of efficiency of thermal conduction system size

Extension of study by Takasao et al (ApJ 2015 805 135)

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

1 Higher compression region (ldquoblobrdquo) for stronger thermal

conduction

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Influence of efficiency of thermal conduction system size

Log10 (T2 MK)

Log10 (N109)

vz [170 kms]

[18 s]

1206390 = 0 1206390 = 027 1206390 = 27

β = pg pm = 008

2 Enhanced evaporation for stronger thermal conduction

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Long Duration GOES C67 flare

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Some remarksbull AIA differential emission measure inversions (eg using the method of

Cheung et al 2015 httptinyurlcomaiadem) can be used to quantitatively study the thermal evolution of flare plasma

bull The efficiency of thermal conduction system size impacts

1The density enhancement in deceleration region of the reconnection outflow and

2The amount of evaporated material

bull Observations can possibly help us test whether classical Spitzer thermal conduction is appropriate or should be modified (eg Imada Murakami amp Watanabe PoP 2015 22 10 Wang et al 2015 ApJL 811 L13)

bull Future work

bull Should measure 1 and 2 in a systematic study of flares to test sensitivity to system size efficiency of thermal conduction

bull Use 3D MHD simulations of flares for forward modeling of observables

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

Science Case 3) Chromospheric Evaporation amp Condensation

IRIS is designed to probe the chromosphere and transition region but it also sees flare plasma (Fe XXI 10 MK) W h a t a b o u t t h e temperature range in between Join t observat ions between IRIS andAIA to fill the temperature gap

Science ObjectivesThe IRIS science investigation is centered on 3 themes of broad significance to solar and plasma physics space weather and astrophysics aiming to understand how internal convective flows power atmospheric activity

1 Which types of non-thermal energy dominate in the chromosphere and beyond

2 How does the chromosphere regulate mass and energy supply to corona and heliosphere

3 How do magnetic flux and matter rise through the lower atmosphere and what role does flux emergence play in flares and mass ejections

Observations Spectra covering temperatures from 4500 K to 10 MK

Images covering temperatures from 4500 K to 65000 K

Baseline 5s for slit-jaw images 1s for 6 spectral windows rapid rastering

Predicted Count Rates

Observing Modes

OverviewThe primary goal of NASArsquos Interface Region Imaging Spectrograph (IRIS) small explorer is to understand how the solar atmosphere is energized The IRIS investigation combines advanced numerical modeling with a high resolution UV imaging spectrograph

IRIS will obtain UV spectra and images with high resolution in space (13 arcsec) and time (1s) focused on the chromosphere and transition region of the Sun a complex dynamic interface region between the photosphere and

corona In this region all but a few percent of the non-radiat ive energy l e a v i n g t h e S u n i s converted into heat and radiation IRIS fills a crucial gap in our ability to advance Sun-Earth connection studies by tracing the flow of energy and plasma through this foundation of the corona and heliosphere

General

Multi-channel imaging spectrograph with 20 cm UV telescope

Spectra along a slit (13 arcsec wide) and slit-jaw images

CCD detectors with 16 arcsec pixels

Effective spatial resolution 033 (FUV) and 04 arcsec (NUV)

Maximum field of view 120x120 arcsec

Spectra Far-UV channels 1332-1358Aring amp 1390-1406Aring at 40 mAring resolution

Near-UV channel 2785-2835Aring at 80 mAring resolution

Slit-jaw Images 1335 Aring and 1400 Aring with 40 Aring bandpass each

2796 Aring and 2831 Aring with 4 Aring bandpass each

Instrument20 cm UV telescope + spectrograph

Mission DetailsSun-synchronous polar orbit continuous observations 8 monthsyear

Expected launch date around December 2012

Data rate 07 Mbitss 3-60 more than previous imaging spectrographs

Coordinated observations with Hinode SDO STEREO amp ground-based observatories for photospheric magnetograms and coronal imaging

Collaborating Institutions LMSAL (PI Alan Title)

LMSampES (spacecraft) NASA ARC (mission ops)

SAO (telescope) MSU (spectrograph)

LSJU (data handling) UiO (modeling data)

High throughput allows for rapid rasters of high SN spectra that allow line centroid velocity determination down to 05 kms precision within 1 s exposures for brightest lines

Numerical ModelingThe IRIS science investigation has a strong theorynumerical modeling component State-of-the-art radiative 3D MHD numerical simulations and synthetic (non-LTE) diagnostics in eg optically thick lines like Mg II hk will allow de ta i l ed compar i sons w i th IR I S observables Such comparisons are key to interpreting the IRIS data and ultimately determining the non-thermal energization of the solar atmosphere

Comparison to SUMEREIS

Example showing synthetic slit-jaw images and spectral profiles in Mg II hk by using MULTI_3D on snapshots of 3D radiative MHD simulations from the University of Oslo (BIFROST) Courtesy Mats Carlsson (UiOITA)

Example showing intensity doppler shift line width and spectral profiles of the optically thin Si IV 1394 A line by using CHIANTI on snapshots from BIFROST simulations Courtesy Viggo Hansteen (UiOITA)

The high throughput amp spatial resolution and simultaneous slit-jaw imag ing o f IR IS wi l l prov ide unprecedented v iews o f the c o n n e c t i o n b e t w e e n t h e chromosphere TR and corona For example IRIS can take a full spectral raster across 6 arcsec and context slit-jaw imaging at 033 arcsec resolution within the time it takes SUMER or EIS to expose one slit pos i t ion (~20s ) a t 2 arcsec resolution Ask to see the movie

IRIS Small ExplorerInterface Region Imaging SpectrographPI Alan Title (Lockheed Martin Solar amp Astrophysics Lab)

httpirislmsalcom

Co-InvestigatorsA Title (PI) W Abbett M Carlsson M Cheung E DeLuca B De Pontieu B Fleck S Freeland L Golub V Hansteen L Harra J Harvey N Hurlburt P Judge J Lemen C Kankelborg D Klumpar B Lites S McIntosh S Poedts P Scherrer C Schrijver R Shine T Tarbell D Tenerelli S Tsuneta H Uitenbroek A van Ballegooijen N Waltham M Weber T Wiegelmann M Wheatland S Worden J-P Wuelser M Yamada

From the IRIS poster irislmsalcom

IRIS SJI 1330 Diffuse long-lived loops reaching into the corona are generally attributed to Fe XXI 1354 Å emission

At httpwwwlmsalcomdatahtml go to lsquoList of Flares Observed with IRISrsquo compiled by Kathy Reeves

20140917 - 1926 C75 flare - behind

the limb nice big loop of Fe XXI visible red shift in

Fe XXI13

IRIS SJI 1330 Likely Fe XXI 1354 Å emissionC II 1336 O I 1356 Si IV 1403 Mg II k 2796 SJI 1330

GOES X-ray

SJI Total DNimage

IRIS SJI 1330 Likely Fe XXI 1354 Å emissionC II 1336 O I 1356 Si IV 1403 Mg II k 2796 SJI 1330

GOES X-ray

SJI Total DNimage

Lower right panel only Greyscale Blos from HMI Green 6MK EM YellowRed 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

Lower right panel only Greyscale Blos from HMI YellowGreen 6MK EM Red 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

Lower right panel only Greyscale Blos from HMI YellowGreen 6MK EM Red 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

IRIS SJI 1330 Coronal condensations appear at about same time (~2029 UT) as when AIA sees sub-MK plasma

IRIS SJI 1330 Likely Fe XXI 1354 Å emissionC II 1336 O I 1356 Si IV 1403 Mg II k 2796 SJI 1330

GOES X-ray

SJI Total DNimage

Lower right panel only Greyscale Blos from HMI Green 6MK EM YellowRed 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

Lower right panel only Greyscale Blos from HMI YellowGreen 6MK EM Red 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

Lower right panel only Greyscale Blos from HMI YellowGreen 6MK EM Red 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

IRIS SJI 1330 Coronal condensations appear at about same time (~2029 UT) as when AIA sees sub-MK plasma

Lower right panel only Greyscale Blos from HMI Green 6MK EM YellowRed 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

Lower right panel only Greyscale Blos from HMI YellowGreen 6MK EM Red 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

Lower right panel only Greyscale Blos from HMI YellowGreen 6MK EM Red 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

IRIS SJI 1330 Coronal condensations appear at about same time (~2029 UT) as when AIA sees sub-MK plasma

Lower right panel only Greyscale Blos from HMI YellowGreen 6MK EM Red 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

Lower right panel only Greyscale Blos from HMI YellowGreen 6MK EM Red 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

IRIS SJI 1330 Coronal condensations appear at about same time (~2029 UT) as when AIA sees sub-MK plasma

Lower right panel only Greyscale Blos from HMI YellowGreen 6MK EM Red 10 MK EM

Other panels EM in various log T bins

bull Tell-tale signs of chromospheric evaporation

bull Loops filled with plasma at 10 MK and above

bull Loops cool to lower log T bins

bull At time (~2029 UT) when plasma cools down to log TK ~ 58 coronal condensations in SJI 1330 begin to appear

IRIS SJI 1330 Coronal condensations appear at about same time (~2029 UT) as when AIA sees sub-MK plasma

IRIS SJI 1330 Coronal condensations appear at about same time (~2029 UT) as when AIA sees sub-MK plasma