Fractal Dimension and Maximum Sunspot Number in Solar Cycle

R.-S. Kim1,2, Y. Yi1, S.-W. Kim2, K.-S. Cho2, Y.-J. Moon2

1Chungnam national University

2Korea Astronomy and Space science Institute

Contents

Introduction

Prediction of solar activity cycle

Fractal dimension

Calculation of the fractal dimension

Cycle variation of the fractal dimension

Prediction of maximum sunspot number

Conclusion

Prediction of solar activity cycle

Prediction of solar activity cycle

When, how strong~?

Prediction method

Period analysis: power spectrum

Gleissberg cycle: long period (80 years)

Precursor method: extended period

Fractal dimension

What is the Fractal?

Fractal has statistical self-similarity at all resolutionsand is generated by an infinitely recursive process.

Fractal dimension

New parameter for quantitatively describing the characteristics of an irregular time series.

Higuchi’s Method (1988)

The fractal dimension approaches

1, if a time series is regular ‘Marginal fractal’

2, if a time series is completely random ‘Extreme fractal’

N: data number, m: initial time, k: interval time (resolution)

Fractal dimension

m

Fractal dimension

The length of the curve for the time interval k, <L(k)>

The average value over k sets of Lm(k)

Lm(k) = { ( ∑ | X(m+ik) - X(m+(i-1) k ) | ) × (N-1) /[(N-m)/k]k}

× 1/k

k ↓, data number ↑ ; k ↑, data number ↓

if | X(m+ik) - X(m+(i-1) k ) | is constant, <L(k)> ~ k-2

Extreme fractal

If <L(k)> ~ k-D, the curve’s fractal dimension is D

Cycle fractal dimension

Sunspot number: The daily relative sunspot numbers since 1850 from SID

C

(Solar Influences Data analysis Center).

log<L(k)> = a – D log k (a = constant)

Cycle variation of the fractal dimension

Cycle variation

10 ~ 22 solar cycle

Cycle variation of the fractal dimension

Cycle variation of the fractal dimension

Sunspot number

Cycle variation of the fractal dimension

Fractal dimension vs. Maximum sunspot number (r = -0.95)

Fractal dimension vs. Maximum sunspot number

The coefficient of correlation for increasing phase

Prediction of maximum sunspot number

Observed vs. predicted maximum sunspot number

Using fractal dimension of increasing phase (4 years)

Prediction of maximum sunspot number

Conclusion

Variation of the fractal dimension

Cycle fractal dimensions and maximum sunspot numbers have inverse relationship.

Prediction method

By using fractal dimension for increasing phase of solar cycle,

we can predict maximum sunspot number

Observed and predicted sunspot numbers have good relationship (r = 0.90).