Diagnosis of Stochastic Fields by Mathematical
Morphology and Computational Topology Methods.
Solar Magnetic Field Radioactive Contamination Seismic Events
Minkowski Functionals
K
Ke
Convex set K in
The parallel set of distance to K is
a closed ball of radius at
x K
K B x,
Evolution of the covering of a set of points
B x, x K
dR
Makarenko N.G., Karimova L.M.
Steiner formula as definition of the Minkowski functionalsK.Michielsen, H.De Raedt. Integral-geometry morphological analysis. Physical Reports v.347, 6, 2001
is dimensional volume.Completeness: Minkowski functionals in space.
-------------------------------------------------------------------------
is edge of square . --------------------------------------------------------------------------
0
di
ii
V K W K
V d
1d dR
2d : 0 1
1
2W K A K , W K L K 2W K
3d :
a K
0 13W K V K , W K F K
V volume, F area
2 33 3 4W K H K , W K G K K
2 24V K a a
A area, L line length, Euler characteristic
H int egral, G Gaussian curvatures
Morphological properties:
• Motion invariance -translation and rotation• Additivity
• Continuity
when
gK K g G
1 2 1 2 1 2K K K K K K
l llim K K
l llim K K =0
Euler characteristic : =#vertices-#edges+#faces is topological and morphological invariant
d = 2, #connectedcomponents #holes
d = 3, #connectedcomponents #tunnels #cavities
Adler R.J., The Geometry of Random Fields, Wiley,New York, 1981
Boulingand-Minkowski Dimension
0
M
logvol Kd K d
loglimsup
• Dilation
• The change of the parallel body volume gives the Minkowski dimension
Sorted Exact Distance Representation Method
Scheme of dilation of the central point
Pattern 1st step of dilation 2nd step of dilation1 2 2 5
L.da F. Costa, L. F. Estrozi, Electronics Letters, v.35, p.1829, 1999
Minkowski Functionals and Comparison of Discrete Samples in Sismology
• Six five-year samples represented by earthquake epicentres in the East Tien Shan (log E>10) • Seismic events in California
• Model of Poisson distribution
74 80 40 46
The functional W0 (area of the covering) versus the radius
Makarenko N.,Karimova L., Terekhov A.,Kardashev A. Izvestiya, Physics of the Solid Earth, 36, No 4,305-309, (2000)
The functional W1 (perimeter of the covering) versus the radius
The functional W2 (Euler characteristic) versus the radius
Minkowski functionals curves
• are different for Tien Shan and California regions
• remain almost unchanged for six five-year intervals
• differ from model of Poisson distribution
Mecke K.R.,Wagner H., J.Statist. Phys., 64, no3/4, 843-850, (1991)
TOPOLOGICAL COMPLEXITY OF RADIOACTIVE CONTAMINATION
Radioactive contamination of Kazakhstan
470 nuclear explosions on Semipalatinsk test site 90 explosions in the air 25 on the ground 355 underground.
Data array of Cs• Measurements along a grid of parallel lines . Karaganda and Semipalatinsk regions km Irtysh area (m) Spectrometer, -quanta flow density (0.25-3.0 Mev)
214Bi (1.12 and 1.76 Mev) ------U 208Tl(2.62 Mev)-------------------Th 40K (1.46 Mev)--------------------K 137Cs(0.66 Mev)-------------------Cs
•Litochemical measurements. Method of soil samples. (m) Irtysh area, 137Cs isotope
Measurements
Irtysh Test Site
0 1 8 9 10 11
0,0
0,1
0,2
0,3
0,4
gapg2 g1g3
ground aero
km
km
Paving map of U isotope, g3 Irtysh area, aerogamma measurements.
0 2000 4000 6000 8000 10000
0
50
100
150
200
250
300
co
nta
min
atio
n
n
Topological classification of radioactive contamination
-6 -4 -2 0 2 4 6
-1.0
-0.5
0.0
0.5
1.0
Cs
aero1 aero2 aero3 g1 g2 g3 aero12 gauss aero12,disc aero3,disc
HA
-4 -2 0 2 4 6
-40
-30
-20
-10
0
10
20
30
40
50
Cs K Th U gauss Cs,g3 K,g3 Th,g3 U,g3
HA
curves for 2 grounds
curves of Cs data
• Morphological characteristics differ from Gauss field one.
• Man-made Cs topology differs from U,Th,K topology
• Shapes of curves are enough robust to the variation of sample volume
Makarenko N.,Karimova L., Terekhov A.,Novak M. Physica A, 289,278-289, (2001)
Computational Topology
0 1
log Nlim inf
log
0,3 0,4 0,5 0,6 0,7 0,8 0,90,4
0,8
1,2
1,6
2,0
2,4 Th K U Cs
Disconnectedness index for Th,K,U,Cs.
N
Disconnectedness index:
is the number of -components of given resolution and
intensity of measure
”Hot spots" of contamination is forming the set of small dimension.
net aD D m Two sets intersect transversely in
mR if
Let is the number of boxes of size with
Probability of finding is
- number of non-empty -boxes.
D - box dimension of the measure support. DN / N p
N ic p
N
Robins V.,Meiss J.D.,Bradley E.,Nonlinearity, 11, 913 ,(1998)
Makarenko N.,Karimova L., Terekhov A., Novak M., Paradigms of Complexity, World Scientific, 269-278, (2000)
• The 11-year period of the sunspot cycle
• The equator-ward drift of the active latitude
• Hale’s polarity law and the 22-year magnetic cycle
• The reversal of the polar magnetic field near the time of cycle maximum
Magnetic Field Charts
Butterfly diagram
SOLAR MAGNETIC FIELD ACTIVITY.
Stanford Photospheric chart 1728 Carrington Rotation
H chart 1700 Carrington Rotation
1600 1700 1800 1900 2000
400
600
800
Carrington Rotations
Are
a SP
erim
ete
r P
400
800
1200
1600 S
Minkowski Functionals for Stenford charts
800 1000 1200 1400 1600 1800 2000
-10
0
10
20
Carrington Rotations
Perimeter P (W0) and area S (W1)
1920 1940 1960 1980 2000-10
0
10
20
years
-200
-100
0
100
200
Wo
lf nu
mb
ers
Wolf
Euler characteristic for 815- 1972 Carrington Rotations
Smoothed and Wolf numbers
Makarenko N.,Karimova L.,Novak M., Emergent Nature, World Scientific, 197-207, (2002)
1600 1650 1700 1750 1800 1850 1900 1950 20000
5
10
15
20
25
30
35
40
Fla
re I
nd
ex
Q
Min
kow
ski dim
en
sion
dMCarrington Rotations
Q
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40 d
1650 1700 1750 1800 1850 1900 1950 2000
0
5
10
15
20 Q
Fla
re in
de
x Q
Carrington Rotations
400
450
500
550
600
650
Pe
rime
ter P
P
• Minkowski Dimension
and Flare Index.
• Smoothed Flare Index and Perimeter.
Coincidence after shifting P on 12 rotations forward.
Interrelation between Large Scale Magnetic Field and Flare Index
x x di j, R
2
1
x xd
i j
number of pairs i, jC
N with
Estimation of Correlation Dimension
Scaling , -correlation dimension dC v
i jx x 2 24jx xi / h
e
For : ν 1 95 0 02. . 0 5. % 018K . bit / rotation 27T rotations
For Wolf numbers: ν 1 73 0 05. . 6% 0 04K . bit / rotation125T rotations
Attractors
Attractor of Wolf numbers
Gaussian Kernel Correlation Integral
Attractor of Euler characteristic
Synchronization of directionally-coupled systems
8 10 12 140
2
4
6
8
10
12
14 Kxy
Kyx
Kij
log
The correlation ratio of interrelation between Euler characteristics (X system) and Wolf numbers (Y system).
• Dominant role of the global magnetic field
Can Driver-Response Relationships be deduced from interdependencies between simultaneously measured time series?
Detecting Interdependencies by Means of Cross Correlation Sums
2
i j
i j
i j i jK
i jxy
y y x x
x x
P. Grassberger, J. Arnhold, K. Lehnerts and C. E. Elger,Physica D, 134, 419,(1999)
G. Lasiene and K. Pyragas, Physica D, 120, 369, (1998)
Self-organizing criticality in dynamics of large scale solar magnetic field.
The fragments of 10 Carrington rotations. H charts.
Changes of the number C() of --disconnected components versus a resolution by computational topology method.
C() for 10 fragments not having pole changes C() for 3 fragments having global field rebuilding.
Robins V.,Meiss J.D.,Bradley E.,Nonlinearity, 11, 913 ,(1998)
Makarenko N.,Makarov V.I.,Topological Complexity of H-alfa maps, abstract, JENAM_2000
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0114-186-7 r=0.07
fg()
Wolf numbers
Large Deviation Multifractal Spectrum. Kernel method.
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
0.0
0.2
0.4
0.6
0.8
1.0 122-161-7 r=0.038
fg()
Euler characteristic
Multifractal spectrum of Wolf numbers.
Multifractal spectrum of Euler characteristic.
Classical methods:
Halsey T.C., Jensen M.H, Kadanoff L.P., Procaccia I.,Shraiman B.I., 1986, Phys.Rev. A, v.33, p.114
Chambra A., Jensen R.V., 1989, Phys.Rev.Lett. v.62, p.1327
J.Levy Vehel, INRIA, France
knk kn n
log I, I int erval
n
0
ng
n
logNf
nlim lim
k k kn n n nN # /
12nn n
kn n
n
N K
is density of
K
Formeasure singularity is
ng
n
logNf
nlim