Б. В. Козелов Полярный геофизический институт КНЦ РАН

1. Б. В. Козелов Полярный геофизический институт КНЦ РАН Суббуревая и буревая активность магнитосферы как отклик на стохастическое воздействие солнечного ветра

2. Approach INPUT “BLACK BOX” OUTPUT Activity indexes Solar wind Magneto-sphere Bz(t), v(t)Bz(t) AE(t), Dst(t)

3. Stochastic differential equation

4. Stochastic differential equation It has recently been shown that fluctuations of global quantities (let us name it X(t)) in certain avalanching and turbulent systems can be described by stochastic differential equations (SDEs) driven by a colored noise term with a diffusion coefficient D = g2[X(t)] depending on X(t). The equation also contains a deterministic drift term f [X(t)] which keeps X(t) within certain limits, and the equation takes the form: Here B(t) could be a Wiener process (Brownian motion), a self-similar Gaussian process (a fractional Brownian motion with Hurst exponent H). [M. Rypdal, K Rypdal, arXiv:0807.3416v1, 2008]

5. Strongly driven Zhang sandpile D[x)] H=0.75 f [x)] Part of a time series for the toppling activity. b) The increments of the signal. c) The conditional variance of the increments given the value of x. d) The conditional mean of increments. [M. Rypdal, K Rypdal, arXiv:0807.3416v1, 2008]

6. 2D turbulence simulation D[x)] f [x)] a) Part of a time series for the kinetic energy. b) The increments of the signal. c) The conditional variance of the increments given the value of X. The inset shows the logarithm of this variance versus X. d) The conditional mean of increments. [M. Rypdal, K Rypdal, arXiv:0807.3416v1, 2008]

7. AE data preparation AE AE/t|t=1 min Compute the drift and diffusion term Removing drift term Normalized by conditional std Compare the rest noise and SW driver

8. Stochastic noise process dB(t) extracted from AE-index 1-min data 1 year f[X] Bursts, intermittency => multifractal structure 6 years D[X] Yearly variations, more visible for years of less activity

9. Application to magnetosphere dynamics Magnetosphere dynamics and solar wind beyond second order statistics. […..] Generalization from self-similar to MF noise is needed.

10. SW data preparation “Transplantation” algorithm has been used to fill data gaps in Bz(t) and v(t). Xp Xn tp t tn 1-min OMNI data

11. Usual problems with MF approach to data set 1. Definition of a measure 2. Quality of estimated features (error bars?)

12. “French” MF approach Authors: S.Jaffard, B. Lashermes, P. Abry, H. Wendt ~ 10 papers Strong mathematical background, detailed description of algorithms, error bars for all quantities.

13. References Lashermes B., S.G. Roux, P. Abry, and S. Jaffard, Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders, Eur. Phys. J. B 61, 201–215, 2008. Wendt H., P.Abry, Bootstrap tests for the time constancy of multifractal attributes, 2007. Wendt H., P.Abry, and S.Jaffard, Bootstrap for Empirical Multifractal Analysis (Bootstrap application to hydrodynamic turbulence), IEEE SIGNAL PROCESSING MAGAZINE, P.38-48, JULY 2007. Jaffard S., “Multifractal formalism for functions, part 2: Selfsimilar functions,” SIAM J. of Math. Anal., vol. 28, no. 4, pp. 971–998, 1997. Jaffard S., B. Lashermes, and P. Abry, “Wavelet leaders in multifractal analysis,” in Wavelet Analysis and Applications, T. Qian, M.I. Vai, X. Yuesheng, Eds. Cambridge, MA: Birkhäuser, 2006, pp. 219–264. Abry P., P. Flandrin, M. Taqqu, and D. Veitch, “Wavelets for the analysis, estimation and synthesis of scaling data,” in Selfsimilar Network Traffic and Performance Evaluation. Spring 2000, Wiley. Jaffard S. , B. Lashermes and P. Abry, Wavelet leaders in multifractal analysis, in Wavelet Analysis and Applications, 2005, University of Macau, China. Efron B., The Jackknife, the Bootstrap, and Other Resampling Plans, Society for Industrial and Applied Mathematics, Philadelphia, 1982.

14. Code overview • Discrete wavelet decomposition, Daubechies wavelets. • Wavelet leaders (WL) used instead of wavelet coefficients. • All MF attributes calculated from WL statistics. • Block bootstrap procedure applied at each scale to estimate error bars. • Double bootstrap procedure for estimation of temporal stability of the MF characteristic. • Simple expansion is possible for forecast.

15. Discrete wavelet coefficients (DWC) The discrete wavelet transform is a decomposition (also called multiresolution analysis) of the function f on the orthogonal basis {j,k}jZ,kZcomposed of discrete wavelets j,k: Integers jZ and kZ index the scale a = 2jand the location x0 = k2j. Wavelets {j,k}jZ,kZare space-shifted and scale-dilated templates of a mother-wavelet 0: and define a basis distributed according to a dyadic basis in the space-scale plane. Every wavelet j,k and then every DWC d(j,k) can be associated to the dyadic interval (j,k) = [2jk, 2j(k + 1)[ which will be usefully used for indexing the DWC: d(j,k) = d().

16. Definition of Wavelet Leaders [H.Wendt, P. Abry, 2007]

17. MF attributes LX(j,k) => SX(j,q)  j (q) => (q) singularity spectrum D(h) Log-cumulants. This method provides estimates of the parameters cp of the Taylor series expansion of (q) for q 0 : If the process under analysis is monofractal then c1 = H 0 and cp = 0 for p > 1. A non-zero value for c2 explicitly establishes the multifractal (vs. monofractal) nature of the data and the parameter c2 (also called the intermittency coefficient) is used to characterize the degree of multifractality. A quadratic approximation of the scaling exponents: (q)  c1q + c2q2/2 (when q  0) corresponds to a quadratic approximation of the singularity spectrum: D(h) 1 + (h−c1)2/2c2 (when hc1)

18. [B. Efron, 1982] [H.Wendt, P. Abry, 2007]

19. Time constancy of multifractal attributes Procedure for obtaining T (left) and T *(right) from the wavelet Leaders {LX(j,k)} of X. "cut", "estimate" and “" stand for cutting a set into M subsets, computing estimates ˆ, and bootstrap resampling. [H.Wendt, P.Abry, 2007]

20. Example of MF features for AE-index (2000 year) ζ(q) D(h) c1 c2 c3 ζ(q) is estimated from linear regression of log Z(q,a) versus log a on the time scales 4-128 minutes. A bootstrapping technique has been employed that allows estimation of, and reduction of, error bars.

21. Example of MF features of vBz(2000 year) ζ(q) D(h) c1 c2 c3 ζ(q) is estimated from linear regression of log Z(q,a) versus log a on the time scales 4-128 minutes. A bootstrapping technique has been employed that allows estimation of, and reduction of, error bars.

22. AEvBz c1 c2 c3 2000 2001 2002 2003 2004 2005 2000 2001 2002 2003 2004 2005

23. Example for Dst – index (1-hour data) c1 D[X] c2 f[X] c3

24. Conclusions • The noise process dB(t) extracted from both AE-index and solar wind E-field time series exhibit intermittency characteristics. • At the substorm temporal scales (4-128 min) the leading Hölder exponent c1 (the Hurst exponent) is ~0.40 in AE-index and ~0.46 in solar wind. • Solar wind is more intermittent than the noise process extracted from AE (|c2| is larger in solar wind time series). • In average, the skewness (c3) of the singularity spectrum (if it is significant?) have opposite signs in AE-index and solar wind. • There is a seasonal variation in the AE-index features. • If a synthetic multifractal process can be constructed that reflects these characteristics, the stochastic equation can provide a model for these time series.