Diagnostics ofFlux Emergence on the Sun

Mark Cheung
Staff Physicist @ Lockheed Martin Solar & Astrophysics Laboratory
Visiting Associate Professor @ National Astronomical Observatory of Japan

Observation+ Experiment

TheoryScientific
Computation

Outline(1) Magnetic Field Diagnostics(2) Thermal Diagnostics(3) D a t a - d r i v e n M o d e l i n g(Again)Magnetic Field Diagnostics of Emerging Flux
Extremely large volume of papers studying
photospheric magnetic observations of flux
emergence.
Ground-based instruments: Leka et al. (1996);
Strous et al. (1996); Lites, Skumanich & Martinez
Pillet (1998); Strous & Zwaan (1999); De Pontieu(2002); Kubo, Shimizu & Lites (2003); Watanabe
et al. (2008, 2011); Guglielmino et al (2010);
Yurchyshyn et al. (2010), Rutten et al. (2013)
Balloons: Flare Genesis Experiment
Bernasconi et al. (2002); Pariat et al (2004)
SUNRISE Guglielmino et al. (2012)
MDI Many many papers
Hinode Centeno et al. (2007); Cheung et al.
(2008); Okamoto et al. (2008); Magara (2008);
Gonzalez & Bello Rubio (2009); Otsuji et al.
(2009, 2011); Ishikawa, Tsuneta & Jurčák (2010),
Shimizu, Ichimoto & Suematsu (2012)
SDO/HMI Centeno et al (2012): Liu & Schuck(2012); Toriumi, Hayashi & Yokoyama (2012,2014); Tarr & Longcope (2012); Cheung &DeRosa (2012), Cheung et al. (2015)

1102

PARIAT ET A

Fig. 1.—(a) Observed horizontal field. The background image is the Bz magnetogram,on the vertical field magnetogram used as an input in the extrapolation. One notes the inc
the arrows give the orientation of the horizontal field, and their length is proportional to th
where the total field is stronger than 200 G.

the active region. Schmieder et al. (2004) have shown thatthe highest loops and the northern loops of the region werebest fitted with a lower parameter: ! ¼ 9:4 ; 10"3 Mm"1 . Thisshows that the hypothesis that the whole region has the same
twist is not perfectly true, which is not surprising, since ! isstrictly constant only along a field line. Schmieder et al. (1996)
had already pointed out the existence of a gradient in the

forc
cou
rea
mo
we
non
to r

Ishikawa, Tsuneta &Continuum
Jurčák (2010)•130s cadence raster scans
using Hinode/SP•S t o k e s I n v e r s i o n(SIRGAUS) with an
Linear Pol. gaussian profile flux tube
embedded in the model
atmosphere.
•Sequence of inversions
shows a small loop risingthrough the first one or twopressure scale heights of
Circular Pol. the atmosphere.
•O n l y p o s s i b l e w i t hextremely stable seeing
and sensitivity (space
mission, A/O).

Supersonic downflows in a flux
cancellation site (Cheung et al.
2008) from Hinode/SP. Perhaps asignature of reconnection down
flows.

Flux emergence in the photosphere•••

Stokes I: dark Doppler signatures of small-scale magnetic loops.
Stokes Q & U: elongated features in the body of the loop.
Stokes V: most of the signal is concentrated at the footprints of the loops.

Slide credit: J. De La Cruz Rodriguez

From Cheung et al. (2008): SOT/NFI observations of dark lanes in emerging flux regions.
See also Strous & Zwaan (1999).

1384

CHEUNG ET AL.

Vol. 687

Fig. 16.— Example of a transient darkening in the simulated emerging flux region (run A; k ¼ 0:1). The top and bottom panels, respectively, show the vertical field
From
Cheungal.¼ 0:1)(2008):
Radiative
MHDat 500
simulation
emerging
flux
reproduces
dark
and the emergent
continuum intensity
nm. The transientof
darkeningcorresponds tothe crestof a granular-scale !-loop
emerging
component(sampledet
at !500
"1
j
j
>
100
G,
and
v
¼
0:5
km
s
).
The
footpoints
of
the
loop
correspondto theatthephotosphere(yellow
contours
indicate
magnetic
upflows
with
I/
h
I
i
<
0:7,
B
hor
z
lanes.
Basically,
upflow
regionswithlowentropy.
opposite-polarity flux concentrations at the ends of the darkening (the green and red contours, respectively, enclose positive and negative flux regions with jBz j # 700 G).

Magnetic bubbles in the photosphere
SST/CRISP - Ca II 8542

SST/CRISP - 6302, full-Stokes

Slide credit: J. De La Cruz Rodriguez

Ortiz et al. (2014)

Magnetic bubbles in the chromosphere
SST/CRISP - 8542

Ortiz et al. (2014); de la Cruz Rodríguez et al. (2015)

From Ortiz et al. (2014):• The emergence is not visible at line center in Ca II 8542, during our series.
• Canopy of fibrils above the emerging flux in the chromosphere.
• Line positions along the wing show similar dark features as in the Fe I lines.
• Observed delay in 8542 relative to Fe I 6302.
• No signal in Stokes Q & U, very weak Stokes V signal.

Slide credit: J. De La Cruz Rodriguez

De la Cruz Rodriguez et al. (2015): non-LTE Stokes inversion of
SST Ca II 8542 Å observations of an emerging magnetic bubble.
log tau = -4.4

log tau = -3.0

log tau = -1.9

log tau = -1.0

log tau = 0.0

Above: Sequence of temperature maps at 1 min cadenceBelow: Sequence of Doppler velocity maps at 1 min cadence
log tau = -4.4

log tau = -3.0

log tau = -1.9

Above: Sequence of Bz maps at 1 min cadenceLeenaarts et al. (2013), IRIS ITN 37
Ca II 854.2 (yellow)

Radiative MHD simulation of AR formation* (M. Rempel)*work done for NASA’s Heliophysics Grand Challenges Research project (LMSAL/NCAR/SAO/BAERI/U Oslo)

Bz @ 1.5 Mm

Log mass density

Bz @ 3.8 Mm

Log internal energy density

Integral Field
It would be an amazing achievement to map theSpectropolarimetry chromospheric field to see reconnection in action.
(e.g. Solar-C, DKIST)IRIS raster scan of an EFR:
Mg II k and Mg II triplet

Scanning from the wing
to the k3 (core), one
sees the transition fromreversed granulation toarch filaments / fibrils.

IRIS raster scan of an EFR:
Mg II k and Mg II triplet

Scanning from the wing
to the k3 (core), one
sees the transition fromreversed granulation toarch filaments / fibrils.

IRIS raster scan of an EFR:
Mg II k and Mg II triplet

Scanning from the wing
to the k3 (core), one
sees the transition fromreversed granulation toarch filaments / fibrils.
Scanning to the Mg II
triplet, one seesdifferent structure in thechromosphere(seemingly lower lying
loops).

Blue wing

Near k2v

Near k2r

Mg II triplet

IRIS Mg II rasters of an EFR @ 330s cadence
Mg II triplet lines are a diagnostics for low chromospheric heating(Pereira et al. 2015)

Wouldn’t	 it	 be	 fantastic	 to	 put	 magnetic	 field	 
vectors	 on	 
these	 images?	 	 

Thermal Diagnostics

Statement of the Problem
The Atmospheric Imaging Assembly (AIA, Lemen et al. 2012;
Boerner et al. 2012) instrument onboard NASA’s Solar
Dynamics Observatory (SDO, Pesnell et al. 2012) is a suite offour normal-incidence reflecting telescopes that image the Sun
in Coronalseven EUV
channels,channels and Arcades
one visible
Mass
Sourcetwo
for UV
Post-Eruption
wavelength channel.

4. EMISSION MEASURE DETERMINATION USING AIA

The aim of this and many other studies is to extract thermal
4.1. theStatementof theinformation about
Sun’s opticallythinproblem
corona using the EUV
Thephysicalcalibratedproperties(i.e. dark-subtracted,
flat-fielded
can be observations.
related to theof opticallythin coronal pand exposure time normalized) count rate yi in the i-th EUV
channel is related to the thermal distribution of coronal plasma
Z
1
by:

yi =

Ki (T ) DEM(T )dT,0where Ki(T) is the temperature response function (see nexttime-normalized
pixel
value
in
the
i-th
AIA
channel
(in
units
of
D
slide):
R1

function (in units of DN cm5 s 1 pixel 1 ) and DEM(T ) =n

From aia_get_response.pro in SSW

The eﬀective areas were obtained from P. Boerner (2009, private

Fe X
Fe IX
Fe IX
Cont.

190.04189.94188.50

6.05
5.85
5.85

0.06
0.06
0.07
-

0.04
-

0.05

0.04

211 Å

Cr IX
Ca XVIFe XVIIFe XIVFe XIIIFe XIIIFe XIII
Fe XI
Fe X
Ni XI
Cont.

210.61
208.60204.67211.32202.04203.83209.62209.78207.45207.92

5.95
6.7
6.6
6.3
6.25
6.25
6.25
6.15
6.05
6.1

0.07
0.11
0.05
0.03
0.08

0.13
0.05
0.05
0.12
0.03
0.04

0.39
0.07
0.05
0.07

0.09
0.07
0.12
0.41

304 Å

He II
He II
Ca XVIII
Si XI
Cont.

303.786
303.781302.19303.33

4.7
4.7
6.85
6.2

0.33
0.66
-

0.32
0.65
-

0.27
0.54
0.11
-

0.29
0.58
0.05
-

335 Å

Al X
Mg VIII
Mg VIII
Si IX
Si VIIIFe XVIFe XIV
Fe X
Cont.

332.79335.23338.98341.95319.84335.41
334.18184.54

6.1
5.9
5.9
6.05
5.95
6.45
6.3
6.05

0.05
0.11
0.11
0.03
0.04
0.13
0.08

0.11
0.06
0.06
0.03
0.04
0.15
0.05

0.86
0.04
-

0.81
0.06

coronalhole, quiet Sun,
activeand flarecommunication),withtheregion
exceptionof plasma
the 171 Å and 335 Åchannels for which updated versions were used obtained from
Table 1.(12Predicted
AIA count rates.
Solarsoft
July, 2010).
Ion

3. Results

λ
Å

T pa
K

Fraction of total emission
CH
QS
AR
FL

Table
lines which941Ålists those
Mg VIIIspectral
94.07
5.9 contribute
0.03 - more
- than 3%
Fe XX
7.0 for- CH, QS,- AR- and flare
0.10
to the total emissionin 93.78
each channel
XVIII
93.93
6.85 -contribution
0.85
plasma. AlsoFeincludedisthe fractional
of0.74the conFe X for any
94.01
6.05 the0.63
0.72 contributes
0.05 tinuum emission
case wherecontinuum
93.47
5.6in a channel.
0.04 - Synthetic
more than 3%FetoVIIIthe total
emission
specFe VIII
93.62
5.6
0.05 tra for each of
the channels
are displayedin Fig. 2 - 8.-For everyCont.
0.11 0.12 0.17 channel each spectrum has been divided by the peak intensityof theWeakerspectrabeen -scaled -by
131strongest
Å O VI spectrum.
129.875.45
0.04have
0.05
Fe XXIII
0.07
factors indicatedin each132.91
figure. 7.15 XXI- 93.930.83
The 94 ÅFechannel
is128.75
expected17.05
to observethe- Fe XVIIIÅFe∼VIII6.85] in
130.945.6
0.30
0.25 AR0.09
flaring regions.
For bothand flaren line [log T[K]
VIII2) the131.24
0.39 0.33
plasma (see Fe
Fig.
dominant5.6
contribution
comes0.13
from -theCont.
0.11 0.20 0.54 0.04
Fe XVIII 93.93 Å line. However, for the CH and QS spectrum (see
Å dominant
Ni XIV contribution
171.37
6.35
Fig. 1712) thecomes- from -the Fe 0.04
X 94.01- Å, line [log T ∼Fe
X
174.536.05 0.03 6.05].
IX
171.075.85 0.95
0.92 1 to
0.80
0.54
In flaringFe
regionsthe131 Å channel
is expected
observeCont.
0.23
e the Fe XX 132.84
Å and Fe XXIII 132.91 Å lines. However, forn the flare
193 Åspectrum
O V (see 192.90
0.03 contribution- comesFig. 3) the5.35
dominant
M
Ca 128.75
XVII
192.85- combined
0.08
from the Fe XXIÅline [log6.75
T ∼ 7.05].
The
conm
CaFe
XIVXX 132.84
193.87Å and
6.55- 132.91- Å lines
0.04 is lesstribution
of
the
Fe
XXIII
Fe XXIV
192.03
7.25 0.81
thanten
percent
of
the
total
emission.
ForCHobservations
ther
Fe XII
195.12
6.2
0.08
0.18
0.17
1
to be 6.2
dominated
Fe VIII0.17
lines [log
Fe is
XIIexpected
193.510.09 by0.19
c 131 Å channel
the dominant
contribution
Fe XIIour simulations,
192.396.2
0.04 0.09
0.08 -forl T ∼ 5.6]. From
Fig. 3) does
Fe VIII0.04
lines, -but
Fe XI (see188.23
6.15come
0.09from0.10- CH and QS plasma
Fe XI contribution
192.83 from
6.15
0.05 0.06- For -thecontinuum
emission.
. with a significant
Fe XI
188.306.15 0.04 0.04 e
Fe X
190.046.05 0.06 0.04 Å2
Fe IX
189.945.85 0.06 m
Fe IX
188.505.85 0.07 Cont.

-

-

0.05

0.04

The count rates are normalised for each channel. Coronal hole(CH), quiet Sun (QS), active region (AR) and flare (FL) plasma.
a
T p corresponds to the log of the temperature of maximum abundance.

From O’Dwyer et al. 2010

1
ed
to
the
physicalproperties
ofopticallythincoronal plasma as an integ
4.1.
Statementoftheproblem(T ) DEM(T )dT,yi =Ki (T ) DEM(T )dT,
ons can be related to the physical propertiesof optically thin coronal plas0Z 11i-th AIA 1channel (in utheexposuretime-normalized
pixel
value
in
the
he i-thyiAIA
channel
(in
units
DN
s
pixel
),
K
(T
)
i
i
=
Ki (T ) DEM(T)dT, of
Z 1
R
5
1
1
1
ature response
function
(in
units
of
DN
cm
s
pixel
)
and
DEM(T
)
1
2
0
pixel and) and DEM(T= ) DEM(T
n
(T
)dz,
where
DEM(
y5i = 1)dT
Ki (T
)dT,
e
0
easure
(in
units
of
cm
K
)
of
the
plasmaalong
aline-of-sight.
n
(T
)
e
1
1
0
1pixel value in the i-th AIA channel (in units of DN s
ized
pixel
),
K
K
) whereoftemperature
theDEM(T)plasmaalong
aline-of-sight.
n
(T
)
is
the
elec
a certain
T
.
Let
the
temperature
range
bedividedintonne
R 1 (inis the differential emission measuree2 units of
5
1
1
nure
oftemperature
DN
spixelpixel)inandDEM(T)ne=ne (in(T)dz
is thedi↵-5time-normalized
valuethethedivided
i-th AIA
channel
units
of
DN
cm
K-1)cmof plasmaalongline-of-sight.
(T)iselectron
0 theetunits
the
range
be
into
n
neighboring
R
1 2
Z
n
1
5
1
1
T
+
T
j
j
X
) of
the
plasmaalong
aDN
line-of-sight.
ne (TT.))istheelectronnumber
onse
function
(in
units
cm
s pixelandDEM(T= 0 nede(Tnumberdensityofof
plasmaat temperatureThe
challengeis) toKi (T )DEM(T)dT,
5 DEM(T)i =ofalong
et theoftemperaturebedividedinto
n neighboring
sothe
that
solve
given
ay
set
EUV
measurements
y. nebins,units
cm forK 1 ) rangeof theplasmaaline-of-sight.
(T
)
is
elect
Tj
j=1
mperature
T
.
Let
the
temperature
range
bedividedintonneighboringbZn theTtemperature
Tj
j + TjLet
range be divided into n neighboring bins,
X
j-thyitemperature
bin K
has
range
T
2
[T
,
T
+
T
).
The
aim
is
to
use
A
Z
j
j
j
n
=
(T
)DEM(T
)dT,
T
+
T
so
that:
i
j
j
K
(T
)DEM(T)dT,
X
ieasure distribution(eitherindi↵erential
or
integralform).
Tjj=1
yi =
Ki (T )DEM(T )dT,
that Ki (T ) = Kij is piecewiseTj constant in each j-th temperature bin, so
j=1
ange T where
2 [Tj ,theTj +to nuseAIAT measurements
to retrj ). The aim
j-th Ttemperaturebinis has
range
∈
[T
,T
+∆T
).
j
j
j
X
in
di↵erential
or
integralform).
Tj ).bin
The
aim
is2to
measurements
toretrieve
raturehas range[Tpiecewise
Tj +AIA
T
).
The
aim
is
to
use
AIA
measurj ,usej
Assumingthat KTi(T)isconstantin
each
temperatureyi =Kij EMj , whereiseconstant
in
each
j-th
temperaturebin,soribution(eitherindi↵erential
or
integralform).
tegralform).
bin, we have:j=1= Kij is piecewise
constant
in
each
j-th
temperature
bin,
so
Z
n
h j-th temperaturebin, soTj + TjXn EM =yi =Kij EMj , where
DEM(T )dT.
X
j
j
yi =
Kij EMj ,Twherej=1
Z

Statement of the Problem

that Ki (T ) = Kij is piecewise constant in each j-th temperature bin, so
range T 2 [Tj , Tj + Tj ). The aim is to nuse AIA measurements to retXin di↵erential or integral form).
yi =Kij EMj , where
wise constant in each j-th temperature bin, so
j=1
Z Tj + TjnXyi =Kij EMj , where EMj =DEM(T )dT.

Statement of the Problem: y = Kx
j=1

Tj

Za Tmatrix
The
above
is
j + Tjequation of the form y = Kx, whereori knowledge about the form of DEM(T ) over the range [Tj , Tj + Tj ), itEM
DEM(T)dT.with
• KK
is(T
an
x piecewisen response
matrix*,eachrowcorresponding to temper
the
j )=m
n thatbeconstant
anddefinea
DEM-weightedi
Tj response function of one AIA channel
temperature
this assumption since such information is not available.
•writteny is an m-tuple
correspondingof AIA count (rates), and
n
be
in
the
followingform
rm of DEM(T ) over the range [Tj , Tj + Tj ), it is in principle possible to• x is an n-tuple with components EMj.

constant and define a DEM-weighted temperature response function. H
y = Kx,information is not available.
The
He
II
line
in
the
304
Å
channelisnotwell-modeledby
CHIANTIng forms an (Warren,m ⇥ n matrix
with157,
components, y is not
an usedm-tuplecorrespondingt
2005, ApJ
147) so it K
is ij
usuallyfor DEM
analysis.
= 6 for
AIA. theUsually
we
want
more
than
6 temperaturebins. For m and
m =So6 mwhen
using
94,
131,
171,
193,
211
and
335
Å
channels)
y = Kx,EMj .< n, the matrix equation y = Kx represents an underdetermined system.
<
n (i.e. moretemperaturebins), Eq.to(5)is underdetermined.
mponents
Kij , ythan
is an6 m-tuple
corresponding
measurements
by the A
measureinversions
far,
researchers
havetacklet*Matrix
elements
dependsfunctionsusedforthe com
131,
171,
193, 211andandthus335on
Å basischannels)andx isattempted
an computing
n-tupletowith
Thatis, the DEM solution is chosen to be one such that
integral✓◆mperature bins), Eq. (5) is underdetermined.
This
is
a
well-known
pro
m
2
X

Usual Approach: 𝛘-squared Minimization
Function to minimize: | y - Kx |2 or |(y - Kx)/𝞂|2
Basically, minimize difference between observed and predicted
counts. The benefits of a least-squares approach is that it leads toEuler-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 𝛘-squared minimization is ideal. However, forunderdetermined systems such an approach can be subject to theperils of overfitting.
Usual way to get around this:Parameterization: e.g. Guennou et al (2012a,b),
xrt_dem_iterative2.pro (M. Weber in SSW, see also Cheng et al
2012)
Regularization: e.g. Hannah & Kontar (2012), Plowman et al. (2013)

K~x max(~y ~⌘ , 0),0.
Cheung et al. 2015: The ~xSparse
Solution

13) andprovidesometoleranceforthesolutiontodeviatef
We(14)
address
the inverse
problem usinganapproachdifferentpose ⌘jthan= 2chi-squared
yj . Other
than the The
problemstated satisfyingby (13) - (14minimization.
set ofas
solutionsthe underdetermined matrix equation y = Kx lies in an affine
are imposed.
n#subspace of R .
such that:minimize

We pick the solution x within this subspacenX

xj subject to K~x = ~y , ~x0.

jThe linear program above finds a solution that minimizes the
L1-norm of the nsolution vector (c.f. Candes 2006). This is notXchi-squared minimization.
minimizex
subject
to
K~
x

~
y
+
~
⌘
,
,
~
x
0,
j
If K is the response function sampled by Dirac Delta functions atjspecific temperatures,
this is equivalent to minimizing the total
EM. If other basis functions are K~
used,
there yis no
simplex max(~~⌘ , 0).
corresponding physical interpretation.

The Sparse Solution
We are unaware of physical principles pertaining tocoronal plasma that motivate the optimization problem
posed above. However, this choice has some important
benefits:
1)It does not overfit (consistent with the principle ofparsimony, i.e. Ockham’s 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 (c.f.
Candes & Tao 2006).
BTW the L1-norm of a vector x = 𝝨 |xi|
4) Speed: O(104) solutions / sec with single IDL thread.
See Asensio Ramos & De La Cruz Rodriguez (2015) for application of related
techniques to 2D coupled Stokes inversion.

3) and (14)providesometoleranceforthesolutiontodeviatefromsatpand (14) provide some tolerance for the solution to deviate
itions(13)ose ⌘j = 2 yj . Otherp than the problem as stated by (13) - (14), no ot
we⌘j = 2 yj . Other than the problem as stated by (13) - (14are choose
imposed.
M curve) are imposed. Handling noisenXnXminimizex
subject
to
K~
x
=
~
y
,
~
x
0.
j
In practice,minimize
measurement
uncertainties
imply
that0.thexsubject
to
K~
x
=
~
y
,
~
x
j
j
equality y = Kx may jnot be satisfied. So our method
solves the followed modified linear program:nXnXminimize
xj subject to K~x  ~y + ~⌘ ,minimize
xj subject to K~x  ~y + ~⌘ ,

j

j

~x

0, K~x max(~y ~⌘ , 0).
~x 0, K~x max(~y~⌘ , 0).

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(yη, 0).

𝛘-squared
minimization

Sparse inversion

94

94

335

131

335

131

211

171

vs.
211

171
193

193

Crossmarks are observed counts
Subspace of allowed solutions in our method

Basis Functions for DEM
APPENDIXQUADRATURE SCHEME

Let i = 1, 2, ..., m denote the index over a set of wavelength band channels and/or line spectra. Let the DEM function
be written in terms of a set of positive semidefinite basis functions {bj (log T ) 0 | k = 1, 2, ..., l}, viz.
DEM(log T ) =lXbk (log T )xk ,

(A1)

k=1

with quadrature coefficients xk

0. Approximating the integrals in equation (1) as sums in log T space, we have
yi =

n X
l
X

Kij Bjk xk

log T,(A2)

j=1 k=1

where j = 1, 2, ..., n is the index over temperature bins, Kij = Ki (log Tj ) and Bjk = bk (log Tj ). The response matrix
K = (Kij ) has dimensions m ⇥ n. The basis matrix B = (Bjk ) has dimensions n ⇥ l, with the k-th column vector
corresponding to the k-th basis function bk (log Tj ). Defining the dictionary matrix D = KB, the set of integral
equations (1) can be written in matrix form:~y = D~x,(A3)

Temperature Bins

In	 practice	 we	 solve	 this	 ➜

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 (i.e. l > m), the linear system Eq. (A3) is underdetermined.
For the results in this paper, we use an equidistant grid in log T with log T = 0.1 ranging from log T = 5.5 to 7.5(i.e. n = 21). Over this temperature grid, the set of Dirac-delta basis functions {bDirac| k = 1, ..., n} iskbDirac(log Tj ) = 1, if log Tj = log Tk ,k
= 0, otherwise.

(A4)
(A5)

Recall that the basis matrix B consists of column vectors corresponding to basis functions. So for the set of Dirac-delta
functions BDirac = I (the identity matrix).
In addition to Dirac-delta functions, we also use basis functions consisting of truncated Gaussians. Each Gaussianfunction of width a generates a set of basis functions {bak | k = 1, ..., n}, where(log Tj log Tk )2abk (log Tj ) = exp, if | log Tj log Tk |  1.8a.
(A6)a2= 0, otherwise.
(A7)
The Gaussian basis functions are truncated (i.e. set to zero) for values of log Tj outside the temperature grid used
for inversions. The corresponding basis matrix for this set is denoted Ba . Di↵erent sets of basis functions can be
combined by concatenating their associated basis matrices. For the inversions shown here, we use the combined basis

Validation Exercise 1: Gaussian DEMs

, 203:26 (14pp),
December Journal Supplement Series, 203:26 (14pp), 2012 December
Guennou et al.
The 2012
Astrophysical

(a)

(e)

(a)

(e)
(d)

(d)

(c)

Guennou et al

(b)

(f)

(f)

(g)

(g)

(c)

Guennou et al (2012) reported that when the input gaussian is
P = 0.3 log T . Many perturbations, already visible in Figure 4, are amplified: the distributio
P = 0.1toFigureξ P5. =Same
Figurea4,narrow
but witha plasma
DEM widthsian DEM plasmaξ gauashaving
thermaldistributionof σincreased
logσTe,TbyMonte
e 5000 panel),e , obtainedmoderately
wide
(0.3
log
right
AIA
6-channelIP
wider,
irregular, anddiagonalsolutions.
structure disappeared(seemap
panel
(a)).
The
presenceofmultiplesolutionsofcomparable
probabilities is increased for a large rang
ors nb and sbis, andinvestigating
thetheisothermal(a) Probability
P (T
|T
),
verticallyreading.
The
c
c
P
I
P
I
P
I
I
of plasmathe
temperatures
Tc ,measureleading tovery
different
estimated
Tc from
the ofinput
The probability
is presented whatever
total emissionEM. ((b)and (c))Probability
profilesTc Tforcentral map P (Tc |Tc ) can help us to properly interpret the inversioc .plasma
inversions
yield
spurious
temperatures.
I
P
result, taking into account the secondary solutions
and providing
their respective probability.

al lines in panel(a)). (d) Total probability of obtaining Tc whatever Tc (see Section 1 and Section 2.1 of Paper I).
I

PInequality constraints (10) ande for the solution to deviate
he inequality constraints (10)rance for the solution to de2). Note that this is not thencertainty in each channel isting statistics; in addition toof statistical errors, there are
ad noise) and linear (calibrathe signal. It is not necessary
rive the tolerance ~⌘ on a rige error terms for each obserrance is not used as an error
uced 2 .

Test DEMs:

sian functions in log T space:
EM0(log T log Tc )2⇠(T, Tc , ) = p exp
,
2
2
2⇡

EM0 = 1029 cm2(13)where Tc is the peak temperature and is the Gaussian
width. The normalization is chosensuch that the total
R1emission measure is EM0 = 0 ⇠dlogT .2 The validation test for the inversion method was performed over a
2 Strictly speaking, ⇠(T ) is not the di↵erential emission measure

as defined in Eq. (1). The two are related by the following relation
DEM(T ) = ln 10T 1 ⇠(T ). Nevertheless, we will follow the common practice in the literature and refer to both DEM(T ) and ⇠(T )as di↵erential emission measure functions.

Validation Exercise 2: Quasi-steadyloops in a NLFFF model of AR 11158

Magnetic Model: Quasi-Grad-Rubin Non-linear forcefree Field reconstruction of AR 11158.
Thermal Model: Quasi-steady loops with different
heating functions.

AR 11158 Model A

AR 11158 Model B

Validation Exercise 3: MHD Model

MHD model of AR formation (with thermal conduction) by
Chen et al. (2014A&A…564A..12C, 2015NatPh..11..492C)

Validation Exercise 3: MHD Model

MHD model of AR formation (with thermal conduction) by
Chen et al. (2014A&A…564A..12C, 2015NatPh..11..492C)

CHAPTER 2. X-RAY TELESCOPE SOFTWARE GUIDE*
Validation Exercise 4: AIA-XRT Cross-Comparison*Not discussed in the paper

AIA-XRT Cross-Comparison Analysis Procedure
1.Load and prep an XRT image.
2.Cutout AIA 94, 131, 171, 193, 211 & 335 level 1.5 data
for XRT FOV as indicated by FITS keywords.
1. DEM inversion (sun_coronal_ext.abund, chianti.ioneq).
2. Sample DEMs onto XRT plate scale.
3. Synthesize XRT image by folding response function

against AIA DEM.
3.Use tr_get_disp to align actual and synthetic XRT
images.
4.Compare (following slides).

2D histograms

2D histograms

2D histograms

Synthetic XRT Images

Validation Exercise 4: AIA-XRT Cross-Comparison•

Synthetic XRT images from DEMs derived using
only AIA Reproduces morphology, but counts
are too low compared to XRT by ~ 30 - 40%.

•Is this good enough?

•Is the discrepancy between synthetic and real
XRT images due to limitations of the inversion, or
due to uncertainties of the absolute calibration ofboth instruments?

Log-normal DEMs: AIA 6 channels only

Log-normal DEMs: AIA 6 channels + XRT Be-thinDEM movieof the emergenceof AR 11158

Black pixels arewhere no solutions
were foundlog T = 6.3

log T = 6.5

log T = 6.7

log T = 6.9

log T = 7.1

log T = 7.3

Warren, Brooks & Winebarger (2011)

Side benefit: Image Denoisingy (AIA 131 Level 1.5)

Kx# (from inversion)Data-Driven Modeling (Again)Modeling of Homologous Jets

IRIS• SOT Ca II H images show jet••

structure similar to the jet studied
by Wei Liu et al (2009).
Lower-atmospheric contribution to
Ca II H shows hints of a rotating
pore and (perhaps) flux emergence.
Indirect evidence of flux emergencefrom appearance of elongateddarkenings followed by appearanceof bright grains (e.g. Strous &Zwaan 1999, Guglielmino et al2010).

IRIS Observations• SJI 1400, SJI 1330 and SJI 2796 @ 24s cadence, 60’’ x 60’’ FOV, 0.167” arcsec pixel size
• 20-step NUV and FUV spectral rasters with 2 arcsec steps, 2 min / raster, 38” x 60”
FOV• Focus on FUV spectra in this presentation.

Doppler shift maps using the Si IV 1394 transition region line
(log Tmax = 4.8) observed by IRIS shows helical motion in all
four jets

Red: Blos < 0
Blue: Blos > 0
Green: Bt

Origin of
Homologous Helical Jets
• Heyvaerts & Priest (1977):
Reconnection between emerging
flux and ambient field for solar
flares.
• Shibata et al (1992, 1994):
Reconnection model for jets ->interpret post-jet loop as small
flare.
• Pariat, Antiochos & DeVore (2010)
showed that persistent twisting of aparasitic polarity embedded in
opposite field region can lead torecurrent, homologous jets. N ~ 1
turn is needed twist up the field
before a helical jet is emitted.

Magnetofriction model of homologous helical jets driven
by HMI vector magnetograms (Cheung et al. 2015).

In a nutshell, the data and model
support the mechanism proposed by
Pariat et al. (2009, 2010) for
homologous jets. Helical jets emitted
after about injection of one turn.

Magnetofriction model of homologous helical jets driven
by HMI vector magnetograms (Cheung et al. 2015).

Observation+ Experiment

Physics &Diagnostics
TheoryScientific
Computation

(IRIS), Hinode, STEREO missions, and NSF’s CoMP and DST/IBIS instruments. The goals of
NASA
Grand
Challenges
Research:
this proposal
are Heliophysics
directly aligned with thescience goalsof these missions
and instruments.

Physics
andDiagnostics
oftheDriversof
Solar
Eruptions
3. PROJECT ELEMENTS
Bifrost simulations

Synthesize
Coronal Observables

Compare field evolution between models

Synthesize
Chromospheric
Observables
initial andboundary conditions

Synthesize
Photospheric
Observables

Evaluate
diagnostics
as probes todrivers of
eruptions

MURaM simulations

Stokes IQUVInversion
initial andboundary conditions

Magnetofrictional
simulations

3

NLFFF
Extrapolations

3

3D FieldReconstructions

3

67
Figure 1: Schematic representation of science
investigation Tasks S1-S7. Radiative MHD

HGCR
TEAM
PI: Cheung

RempelMcIntosh

Testa

Malanushenko

Sainz Dalda

De Pontieu

Martinez-Sykora

DeRosaCarlsson

Hansteen

GudiksenChen

68

“Best Observed XFlare”
Sunquake: Judge etal. (2014)
Filament Eruption
before X-flare:
Kleint et al. (2015)
IRIS Fe XXI spectra:
Young et al. (2015)
ChromosphericEvaporation: Li etal. (2015)
69

“Best Observed XFlare”
Sunquake: Judge etal. (2014)
Filament Eruption
before X-flare:
Kleint et al. (2015)
IRIS Fe XXI spectra:
Young et al. (2015)
ChromosphericEvaporation: Li etal. (2015)70

Target AR: Level 0 Model

Magnetofrictional model of a twisted flux tube emerging next
to a single preexisting sunspot. Red and blue show opposite
polarity. Magnetogram at 3 different heights (left most is z=0).

71

Target AR: Level 1 Model

Magnetofrictional modelof a twisted flux tube
emerging next to the‘preceding’ spot of abipolar active region.
Magnetogram at z=0
shown in greyscale.
Field lines illuminated byfield-line-averaged j2.

72

Target AR: Level 2 Model

MURaM-driven Magnetofrictional model
Initial condition is a pair of developed spots. Twisted tube emerged
north of preceding spot. The MURaM calculation stops at
photosphere. Use electric field at z=0 to drive the magnetofriction(MF) model. The MF model indicates there will likely be interesting
dynamics.
Level 3: The MURaM run will be repeated to include the corona.
Level 4: Then we will use Bifrost to get realistic thermodynamic
structures to compute diagnostics.
73Summary•

Magnetic Field Diagnostics: We really learned a lot fromphotospheric measurements. The prime target now should be
the chromosphere (and maybe corona), which is arguably
the best place for magnetic imaging of reconnection in thesolar and astrophysical domains.

•

Thermal Diagnostics: SDO, IRIS and Hinode provide a wealthof information that covers the full range of temperatures in thesolar atmosphere. AIA DEMs can be routinely computed withgood(fair?) match with XRT observations.

•

Data-Driven models tighten the link between theory andobservations. We have many idealized models but it’s time totest them against real cases. It’s also time to back up
qualitative interpretations of observations with modeling. I
hope more students and postdocs think about the problems
that need to be tackled to advance in this direction.
74

Observation+ Experiment

Physics &Diagnostics
TheoryScientific
Computation

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