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AR Persistence

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Learning dynamical regimes of
Solar Active Region via homology
estimation
Irina Knyazeva, Nikolay Makarenko
Central Astronomical Observatory at RAS,
St. Petersburg, Russia
E-mail: [email protected]
CHAOS 2014, 7-10 June 2014, Lisbon, Portugal
Solar Flares
A solar flare is a sudden,rapid and intence
variation in brightness observed over the Sun's
surface or the solar limb. The amount of energy
released is the equivalent of millions of 100megaton hydrogen bombs exploding at the
same time
• CME's particles can collide with crucial electronics
onboard a satellite and disrupt its systems. 98%
sattelite failure caused by solar flares
• Solar particles can interact with Earth's magnetic field
to produce powerful electromagnetic fluctuations.
• Solar flares can temporarily alter the upper
atmosphere creating disruptions to global positioning
systems (GPS)
Solar space missions
• Solar and Heliospheric
Observatory (SOHO) (1996-2010) MDI on
board for magnetic field observation
• Solar Dynamics Observatory (SDO) (2010 -now)
HMI on board - extension of SOHO mission
The goal of the SDO is to understand the influence of the Sun on the Earth and
near-Earth space. The Helioseismic and Magnetic Imager (HMI) is an
instrument designed to study oscillations and the magnetic field at the solar
surface, or photosphere. HMI will provide the first full-disk continuous
observations of solar magnetic fields in all orientations. Prior measurements
(e.g. MDI) measured only the component of the field along the line of sight
to the observer.
Solar magnetogram
MDI:
Full disc 1024X1024
Resolution 2’’/px ~1700 km,
Noise level 20Gs
Magnetograms show "line-of-sight"
magnetic fields
The darkest areas are regions of "south"
magnetic polarity
and the whiter regions "north" polarity.
HMI:
Full disk 4096x4096px
Resolution 0.5’’/px ~425 km,
Noise level 6Gs
• Solar flares are caused by sudden changes of
strong magnetic fields.But there are no formal
criterion.
• Current methods of flare prediction are
problematic, and there is no certain indication
that an active region on the Sun will produce a
flare.
• Analysis of the evolution of the magnetic field
topology in AR key to understanding processes
preceding the Solar Flares.
AR 11429
(big AR with
the series
of flares,
biggest X5.4)
Strength of Flares
(Watt/Sq.m):
A
B
C
M
X
< 10в€’7
10в€’7 - 10в€’6
10в€’6 - 10в€’5
10в€’5 - 10в€’4
10в€’4 - 10в€’3
AR 11520
Biggest AR in 24
cycle, but produce
only several M
flares, and one
X1.4
There are no similar people,
there are no similar AR.
How describe changes in AR?
Morphological approach
HMI magnetogramm I : x п‚® R , x пѓЋ Z п‚ґ Z
is digital functions
Excursion set
пЃ»
Ah ( I ) пЂЅ x I пЂЁ x пЂ© п‚і h
пЃЅ
R.Adler The Geometry of Random Fields
Consider each magnetogram as a
composition of excursion sets.
How to describe topology of binary
set?
Main morphological characteristic is
Euler number . It is the number of
islands minus the number of holes
Morphological approach
If we choose the specific level
we could trace evolution of Euler
for the sequence of magnetogram
пЃЈ
-208,3
Fl
0
-20
250
-40
-80
150
-100
100
-120
-140
50
-160
-180
0
1
3
5
7
9
11
March 2012
B
200
-60
Topological persistence
H. Edelsbrunner, D. Letscher and A. Zomorodian. Topological
persistence and simplification. Discrete Comput. Geom., 28:511-533, 2002.
Persistence: level sets
Study the change in topology
as we sweep from highest
to lowest function values
Pratyush Pranav. Persistent Holes in the Universe. Institute for Maths and its
Applications, Minneapolis 14th Oct, 2013
Persistence: level sets
Topology changes only
at critical points
(maxima, minima,
saddle points):
birth and death process
Persistence diagram (Death(D), Birth (B))
D:D-B
B:B+D
Typical persistence diagram for AR
2500
1400
1200
1500
Death - Birth
Death - Birth
2000
Component Persistence
1000
1000
800
Holes Persistence
600
400
500
200
0
-1000
-500
0
500
(Death+Birth)/2
1000
1500
0
-1000
-500
0
500
(Death+Birth)/2
1000
1500
How to track evolution of AR?
We need to choose specific level of excursion set
in morphological approach, and track evolution at
this level. How to choose the level????
Euler characteristic of persistence diagram
пЃЈ ( PH ) пЂЅ
 l ( B0 )  
l ( B1 )
As a result we have only one!
Characteristic for the
magnetogramm.
Easy to track evolution!
Omer Bobrowski . Algebraic Topology of Random Fields and
Complexes. PhD 2012
AR 11429
Growth with the first flares and
depression before the biggest Flare
пЃЈ0
-208,3
-20
Depression before the
Strongest Flares
пЃЈ
1000
Fl
-83,3
83,3
250
Fl
250
500
-40
200
-60
-80
150
-100
-120
-140
1
2
3
4
5
6
7
8
9
100
50
-1500
50
0
10 11
March 2012
-2000
Fl
110000
0
1
600
B0
B1
150
-500
-1000
Growth before
theFl Flares
120000
200
0
100
-160
-180
Flares index
500
2
3
4
5
6
7
8
9 10 11
March 2012
Growth with the first flares and
depression before the biggest Flare
5000
B0-B1 600
пЃЈ
Fl
0
500
-5000
100000
400
90000
80000
300
-10000
400
-15000
300
-20000
70000
200
60000
50000
100
40000
0
1
2
3
4
5
6
7
8
9
10
-25000
200
-30000
100
-35000
-40000
0
11
March 2012
1
2
3
4
5
6
7
8
9
10 11
March 2012
AR 11158
Depression before the
Set of Flares
пЃЈ
No obvious dynamics
-291,7
250
-20
Fl
200
-30
пЃЈ
-41,7
41,7
200
200
Fl
150
100
-40
150
100
0
-50
100
-100
-60
50
-70
50
-200
0
-80
12
13
14
15
16
Growth before
the Flares
пЃЈ
0
17
18
19
February 2011
B0
B1
110000
100000
250
Fl
200
90000
12
13
14
15
16
17
18 19
February 2011
Depression before the
Set of Flares
-15000
пЃЈ
250
B0-B1
-20000
Fl
200
80000
150
-25000
-30000
70000
100
60000
150
-35000
100
50000
50
40000
-45000
30000
12
-40000
13
14
15
16
17
0
18
19
February 2011
50
-50000
0
12
13
14
15
16
17
18
19
February 2011
AR 11520
Depression before the
Set of Flares
-500,0
пЃЈ
-50
-375,0
Fl
150
Depression before the
Strongest Flares
-41,7
300 пЃЈ
41,7
200
-100
Fl
150
100
-150
100
0
100
-100
-200
50
-250
-200
50
-300
-300
-400
6
7
8
0
9 10 11 12 13 14 15 16 17 18
July 2012
Growth before
the
Flares
150000
Fl
B0
B1
140000
150
130000
-500
6
7
8
0
9 10 11 12 13 14 15 16 17 18
Depression before the
Strongest Flares
пЃЈ
July 2012
B0-B1
Fl
20000
150
15000
120000
100
110000
10000
100
100000
5000
90000
50
80000
0
50
-5000
70000
60000
6
8
10
12
14
0
16
18
July 2012
-10000
-15000
0
6
7
8
9 10 11 12 13 14 15 16 17 18
July 2012
Conclusion
• Changes in evolution of AR could be
traced with methods of computational
topology and morphology
• Topological descriptors don’t depend from
the scale unlike in morphological approach
• Betti numbers can be the basis of
forecasting of Solar flares
Thank You!
Algorithm for computing Betti
number (Find-Union structure)
Make filtration F(s1), F(s2), …, F(sm),
for Holes use backward filtration
with dual graph
B0 = 0;
For i = 0 to n-1 do
if si is a vertex then
B0=B0+1
elseif si is an edge
then
if(Find(u)!=Find(v))
then
B0=B0-1;
Union(u,v)
The dual graph of a plane
graph G is a graph that has a
vertex corresponding to each
B1 = 0;
face of G, and an edge
For i = 0 to n-1 do
if si is a triangle then joining two neighboring faces
for each edge in G
B1=B1+1
elseif si is an dual
edge then
if(Find(u)!=Find(v))
then
B1=B1-1;
Union(u,v)
Cecil Jose A. Delfinado, Herbert Edelsbrunner. An incremental algorithm for Betti
numbers of simplicial complexes. Computer Aided Geometric Design 12(7):771-784
(1995)
Several steps of filtration by digital
image
Holes creation: inverse filtration
Filtration: 4, 14, 4-14, 20, 34, 20-34, …..
B0:
1, 2,
1, 2, 3,
2
Chech Filtration
Chech Filtration
Homology of simplicial complexes
• Topological invariants: Simpical homology
or Betti number
• Number of connected
components (B0)
• Number of holes (or cycles) B1
B0=1, B1=1
A. Hatcher, Algebraic Topology, Cambridge University Press 2002.
Filtration
Filtration is a sequence of subcomplexes such that
1) пѓ† пЂЅ K
2) K
i пЂ«1
0
1
пѓЊ K пѓЊ .... пѓЊ K
пЂЅ K пѓ€пЃі
i
i пЂ«1
, пЃі
i пЂ«1
m
пЂЅ K
пЂ­ sim plex
Filtration associated to a
function
• f a real valued function defined on
the vertices of K
пЃі пЂЅ { v1 ,
, v k } пѓЋ K , f (пЃі ) пЂЅ m ax i пЂЅ1.... k f ( v i )
• The simplices of K are ordered
according increasing f values (and
dimension in case of equal values
on different simplices).
Algebraic topology: simplices
Working with simplex:
Combination of simplices – simplical complexes
Homology and topological persistence. ENS Lyon January 2010. F. Chazal.
Geometrica Group. INRIA Saclay.
Homology by magnetogram
4000
B0 at 11.02
B1 at 11.02
Euler at 11.02
3000
2000
2000
B0 at 15.02
B1 at 15.02
Euler at 15.02
1000
1500
0
-1000
1000
-2000
500
-3000
0
-4000
-400
-300
-200
-100
0
100
200
300
400
500
600
-500
-1000
-1500
-1000
-500
0
500
1000
1500
Автор
iknyazeva
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