Diagnosis of Stochastic Fields by Mathematical
Morphology and Computational Topology Methods.
Makarenko N.G., Karimova L.M.
Solar Magnetic Field
Radioactive Contamination
Seismic Events
Ke
Minkowski Functionals
Convex set K in
Rd
K
The parallel set of distance
K 
xK

to K is
B  x,  
B  x ,   a closed ball of radius  at x  K
Evolution of the covering of a set of points
Steiner formula as definition of the Minkowski functionals
K.Michielsen, H.De Raedt. Integral-geometry morphological analysis. Physical Reports v.347, 6, 2001
V  K  
d
i
W
K

 i 
i 0
V is
d  dimensional volume.
d
Completeness: d  1 Minkowski functionals in R space.
-------------------------------------------------------------------------
d 2:
W 0  K   A  K  , W1  K  
A  area,
1
L K 
2
W 2  K   
L  l i ne l engt h,   Eul er charact eri st i c
V  K    a2  4a   2
a
is edge of square K .
--------------------------------------------------------------------------
d 3:
W 0  K   V  K  , 3W1  K   F  K 
V  vol ume,
F  area
3W 2  K   H  K  , 3W 3  K   G  K   4  K 
H  int egral, G  Gaussian curvatures
Morphological properties:
Motion invariance   gK     K 
g  G -translation and rotation
• Additivity
•
  K1  K 2     K1     K 2     K1  K 2 
• Continuity
=0
Euler characteristic :
 is topological and
morphological invariant
l i ml    K l     K 
when l i ml  K l  K
=#vertices-#edges+#faces
d = 2,   # connect ed component s  # holes
d = 3,   # connect ed component s  # t unnels  # cavi t i es
Adler R.J., The Geometry of Random Fields, Wiley,New York, 1981
Boulingand-Minkowski Dimension
l og vol  K   

dM  K   l i m sup d 

l
og



 0
•
Dilation
•
The change of the parallel
body volume gives the Minkowski
dimension
Sorted Exact Distance Representation Method
L.da F. Costa, L. F. Estrozi, Electronics Letters, v.35, p.1829, 1999
Scheme of dilation of the central point
 1
  2  2
  5
Pattern 1st step of dilation
2nd step of dilation
Minkowski Functionals and Comparison of Discrete Samples in Sismology
Makarenko N.,Karimova L., Terekhov A.,Kardashev A.
Izvestiya, Physics of the Solid Earth, 36, No 4,305-309, (2000)
•
Six five-year samples represented by earthquake epicentres
in the East Tien Shan (log E>10)
40    46
74    80
• Seismic events in California
• Model of Poisson distribution
The functional W0
(area of the covering)
versus the radius
Minkowski functionals curves
The functional W1 (perimeter of the covering) versus the radius
• 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)
The functional W2 (Euler characteristic) versus the radius
TOPOLOGICAL COMPLEXITY OF RADIOACTIVE CONTAMINATION
Radioactive contamination of Kazakhstan
contamination
470 nuclear explosions on Semipalatinsk test site
90 explosions in the air
25 on the ground
355 underground.
300
250
200
150
100
50
0
Measurements
• Measurements along a grid of parallel lines .
Karaganda and Semipalatinsk regions D10
km,
Irtysh area (D10 m)
Spectrometer, g -quanta flow density
(0.25-3.0 Mev)
0
2000
4000
6000
8000
10000
n
Data array of Cs
Irtysh Test Site
ground
aero
0,4
0,3
km
0,2
g2
g3
0,1
gap
g1
0,0
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. (D100 m)
Irtysh area, 137Cs isotope
0
1
8
km 9
10
11
Paving map of U isotope, g3 Irtysh area,
aerogamma measurements.
Topological classification of radioactive contamination
 curves for 2 grounds
Cs
K
Th
U
gauss
Cs,g3
K,g3
Th,g3
U,g3
50
40
30
20
10
HA
0
• Morphological characteristics
differ from Gauss field one.
-10
-20
-30
• Man-made Cs topology differs
from U,Th,K topology
-40
-4
-2
0
2
4
6

 curves of Cs data
aero1
aero2
aero3
g1
g2
g3
aero12
gauss
aero12,disc
aero3,disc
1.0
Cs
0.5
HA
• Shapes of  curves are enough
robust to the variation of sample
volume
0.0
-0.5
Makarenko N.,Karimova L., Terekhov A.,
Novak M. Physica A, 289,278-289, (2001)
-1.0
-6
-4
-2
0
2

4
6
Computational Topology
N    is the number of -components
of given resolution  and   D
intensity of measure
Robins V.,Meiss J.D.,Bradley E.,
Nonlinearity, 11, 913 ,(1998)
Disconnectedness index:
l og N   
g  l i m i nf
l og 1  
 0
”Hot spots" of contamination is forming
the set of small dimension.
m
Two sets intersect transversely in R i f
2,4
Th
K
U
Cs
2,0
1,6
g
1,2
D net  D a  m
Let N    is the number of boxes of size 
with   ci   
Probability p of finding  is
N    / N   p     D g
0,8
0,4
0,3
0,4
0,5
0,6
0,7
0,8
 Th,K,U,Cs.
Disconnectedness index for
N  - number of non-empty -boxes.
D - box dimension of the measure support.
0,9
Makarenko N.,Karimova L., Terekhov A., Novak M.,
Paradigms of Complexity, World Scientific, 269-278, (2000)
SOLAR MAGNETIC FIELD ACTIVITY.
Butterfly diagram
•
•
•
•
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
Stanford Photospheric chart
1728 Carrington Rotation
H chart
1700 Carrington Rotation
S
1600
1200
600
800
Area S
Perimeter P
800
Minkowski Functionals
for Stenford charts
Perimeter P (W0) and area S (W1)
400
1600
400
1700
1800
1900
2000
Carrington Rotations
20
10
Euler characteristic 
for 815- 1972 Carrington Rotations

0
-10
800
1000
1200
1400
1600
1800
2000
Carrington Rotations
Wolf

200
Smoothed  and Wolf numbers
20

10
0
0
-100
-10
-200
1920
1940
1960
years
1980
2000
Wolf numbers
100
Makarenko N.,Karimova L.,Novak M.,
Emergent Nature, World Scientific,
197-207, (2002)
Interrelation between Large Scale Magnetic Field and Flare Index
d
Q
1.40
35
1.35
30
1.30
25
1.25
20
1.20
15
1.15
10
1.10
5
1.05
0
1600
1650
1700
1750
1800
1850
1900
1950
1.00
2000
Minkowski dimension dM
Flare Index Q
40
• Minkowski Dimension
and Flare Index.
Carrington Rotations
P
Q
650
20
15
550
10
500
5
450
0
400
1650 1700 1750 1800 1850 1900 1950 2000
Carrington Rotations
Coincidence after shifting P on 12 rotations forward.
Perimeter P
Flare index Q
600
• Smoothed Flare Index
and Perimeter.
Estimation of Correlation Dimension
xi , x j  R
Scaling
d
1 number of pai rs  i , j  
Cd     2  

N
wi t h xi  x j  

Cd      ,
v -correlation dimension
Gaussian Kernel Correlation Integral


   xi  x j

 xi  x j
e
2
/ 4 h2
Attractors
Attractor of
Wolf numbers
ν  1.95  0.02   0.5%
K  018
. bi t / rot at i on T 27 rot at i ons
For :
For Wolf numbers: ν  1.73  0.05
  6%
K  0.04 bi t / rot at i on
T
125 rot at i ons
Attractor of Euler
characteristic
Synchronization of directionally-coupled systems
P. Grassberger, J. Arnhold, K. Lehnerts and C. E. Elger,Physica D, 134, 419,(1999)
Can Driver-Response Relationships be deduced from interdependencies between
simultaneously measured time series?
Detecting Interdependencies by Means of Cross Correlation Sums
G. Lasiene and K. Pyragas, Physica D, 120, 369, (1998)
K xy    
i j
14

y i   y  j     x i   x  j 
2
i j    x  i   x  j  

Kxy
Kyx
12
The correlation ratio
of interrelation between
Euler characteristics (X system)
and Wolf numbers (Y system).
10
8
Kij 6
4
• Dominant role of the global magnetic field
2
0
8
10
log 
12
14
Self-organizing criticality in dynamics of large scale solar magnetic field.
Makarenko N.,Makarov V.I.,Topological Complexity of H-alfa maps, abstract, JENAM_2000
Changes of the number C() of --disconnected components versus a resolution 
by computational topology method. Robins V.,Meiss J.D.,Bradley E.,
Nonlinearity, 11, 913 ,(1998)
The fragments of 10
Carrington rotations.
H charts.
C() for 10 fragments not having pole changes
C() for 3 fragments having global field rebuilding.
Large Deviation Multifractal Spectrum. Kernel method.
J.Levy Vehel, INRIA, France
Wolf numbers
1.0
114-186-7 r=0.07
For measure  si n gulari ty i s
k

l
og

I

 , I k  i nt erval
n
 nk 
n
n
0.8
f g    l i m
0.6
fg()
 0
0.4
lim
n 
l og N n
n
N n  #  nk /  nk       nk   
0.2
0.0
0.0
0.1
0.2
0.3

0.4
0.5
Multifractal spectrum of Wolf numbers.
122-161-7 r=0.038
n  
n i s densi t y of  nk
K
Euler characteristic
1.0
N n    2n 1  K 
n    
l og N n
f g    l i m
n
n 

0.8
0.6
fg() 0.4
Classical methods:
0.2
0.0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45

Multifractal spectrum of Euler characteristic.
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