Diagnosis of Stochastic Fields by Mathematical Morphology and Computational Topology Methods. Makarenko N.G., Karimova L.M. Solar Magnetic Field Radioactive Contamination Seismic Events Minkowski Functionals Convex set K in The parallel set of distance to K is a closed ball of radius at 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 is dimensional volume. Completeness: Minkowski functionals in space. ------------------------------------------------------------------------- is edge of square . --------------------------------------------------------------------------
Motion invariance • -translation and rotation • Additivity • Continuity • when Morphological properties: =0 Euler characteristic : =#vertices-#edges+#faces is topological and morphological invariant Adler R.J., The Geometry of Random Fields, Wiley,New York, 1981
Boulingand-Minkowski Dimension • 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 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) • Seismic events in California • Model of Poisson distribution The functional W0 (area of the covering) 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 The functional W1 (perimeter of the covering) versus the radius 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 contaminationof Kazakhstan 470 nuclear explosions on Semipalatinsk test site 90 explosions in the air 25 on the ground 355 underground. 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) • 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 Data array of Cs Irtysh Test Site Paving map of U isotope, g3 Irtysh area, aerogamma measurements.
curves for 2 grounds Topological classification of radioactive contamination • Morphological characteristics • differ from Gauss field one. • Man-made Cs topology differs • from U,Th,K topology • Shapes of curves areenough • robust to the variation of sample • volume curves of Cs data Makarenko N.,Karimova L., Terekhov A., Novak M. Physica A, 289,278-289, (2001)
is the number of e-components of given resolution e and intensity of measure Robins V.,Meiss J.D.,Bradley E., Nonlinearity, 11, 913 ,(1998) Computational Topology Disconnectedness index: ”Hot spots" of contamination is forming the set of small dimension. Two sets intersect transversely in Let is the number of boxes of size e with Probability of finding a is - number of non-empty -boxes. D - box dimension of the measure support. Makarenko N.,Karimova L., Terekhov A., Novak M., Paradigms of Complexity, World Scientific, 269-278, (2000) Disconnectedness index for Th,K,U,Cs.
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
Minkowski Functionals for Stenford charts Perimeter P (W0) and area S (W1) Euler characteristic for 815- 1972 Carrington Rotations Smoothed and Wolf numbers Makarenko N.,Karimova L.,Novak M., Emergent Nature, World Scientific, 197-207, (2002)
Interrelation between Large Scale Magnetic Field and Flare Index • Minkowski Dimension and Flare Index. • Smoothed Flare Index and Perimeter. Coincidence after shifting P on 12 rotations forward.
Estimation of Correlation Dimension Scaling , -correlation dimension Gaussian Kernel Correlation Integral Attractors Attractor of Wolf numbers For : For Wolf numbers: Attractor of Euler characteristic
P. Grassberger, J. Arnhold, K. Lehnerts and C. E. Elger,Physica D, 134, 419,(1999) Synchronization of directionally-coupled systems 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) • Thecorrelation ratio • of interrelation between • Euler characteristics (X system) • and Wolf numbers (Y system). • Dominant role of the global magnetic field
Makarenko N.,Makarov V.I.,Topological Complexity of H-alfa maps, abstract, JENAM_2000 Self-organizing criticalityin dynamics of large scale solar magnetic field. 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 Multifractal spectrum of Wolf numbers. Classical methods: Halsey T.C., Jensen M.H, Kadanoff L.P., Procaccia I., Shraiman B.I., 1986, Phys.Rev. A, v.33, p.114 Multifractal spectrum of Euler characteristic. Chambra A., Jensen R.V., 1989, Phys.Rev.Lett. v.62, p.1327
