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