Possibility of Predicting Solar Activity Using Fractal Analysis

R. Samuel Selvaraj¹ and S. Tamil Selvi²

¹Department of Physics, Presidency College, Chennai

²Department of Physics, Dhanalakshmi Srinivasan College of Engineering and Technology, Chennai

ABSTRACT:

The study of solar activity and solar terrestrial relations, the sunspot number has always been taken as the main indicator of the intensity of solar activity. Various new techniques like neural networks, learning nonlinear dynamics and others are used by researchers to predict solar activity. But we are yet to obtain reasonably good results. This is mainly because the reason of the variation of solar activity is still unknown. Hence it is important to analyze the characteristics of the data. This paper considers sunspot as the index of solar activity and fractal analysis is used to examine the predictability of solar activity. For the period 1994 to 2008, the average fractal dimension for periods of 10 days or less was about 1.49. But during the same period, the average fractal dimension was 1.92 for periods longer than10 days. Hence the result is encouraging for short-term prediction (i.e.) within about 10 days, but discouraging for medium-term prediction (longer than 10 days ).

KEYWORDS: solar activity, neural networks, nonlinear dynamics, fractal analysis, fractal dimension.

INTRODUCTION:

The historical data of the sunspot index have been attracting researchers for more than a century. The most impressive feature of solar activity is its cyclic character with an 11- year periodicity .¹The variation in sunspot number is a good indicator of the general level of solar activity. “Today the notions of fractals and fractal dimensions are used in all natural sciences, economics, medicine and other disciplines. Owing to their universality and outstanding effectiveness in the study of complicated systems and processes of various natures, these notions and the corresponding methods have gained great popularity in the last few decades”.² “The sunspot numbers find its utility in selection of orbits for satellites, prediction of high frequency propagation, prediction of weather and so on. Thus, the prediction of sunspot numbers gains importance”.³ “The recent techniques such as neural networks”⁴ and “learning nonlinear dynamics”⁵are being used to predict sunspot numbers. But, the mechanism of variation of solar activity and hence the sunspot number is not clearly understood till date. This paper uses the daily sunspot numbers as an index of solar activity and the data is subjected to fractal analysis. The fractal dimension values thus obtained is used as an indicator to examine the predictability of solar activity.

We have used the daily sunspot numbers for the period of fifteen years from 1994 to 2008. The international sunspot number is produced by the Solar Influences Data Analysis Center (SIDC), World Data Center for the Sunspot Index, at the Royal Observatory of Belgium. The relative sunspot number is an index of the activity of the entire visible disk of the sun. It is determined each day without reference to the preceding days.

Each isolated cluster of sunspots is termed a sunspot group, and it may consist of one or a large number of distinct spots whose size can range from 10 or more square degrees of the solar surface down to the limit of resolution (e.g., 1/25 square degree). The relative sunspot number is defined as R = K (10g + s), where g is the number of sunspot groups and s is the total number of distinct spots. The scale factor K (usually less than unity) depends on the observer and is intended to effect the conversion to the scale.

Fractal analysis method for calculating the Fractal Dimension:

Various techniques are available to calculate fractal dimension of the given time series data. Of these Higuchi (1988) developed a new method for calculating the fractal dimension of an irregular time series. This method gives relatively accurate results and hence we use the same to calculate the Fractal Dimension (D). Higuchi’s method is as follows. We consider a finite set of time series taken at a regular interval: X(1), X(2), X(3), ….., X(N) Where N is the total number of observations of the given time series. From the given time series, we construct a new time series, { X(m), X(m+k), X(m+2k), …, X(m+[(N – m) / k]. k), Where [ ] denotes the Gauss’ notation and both k and m ( m = 1,2,3,…,k) are integers; m and k indicate the initial time and the interval time; respectively. Then k sets of new time series are obtained. We define the length of the curve of the new time series as follows:

[(N-m)/k]

L_m (k) = {( ∑ | ( X(m + ik) – X(m + ( i- 1) . k ) | )

i=1

N - 1

_____________ } / k

[(N – m) / k] . k

The term, N -1 / [(N-m) / k]. k represents the normalization factor for the curve length of the subset time series. We define the length of the curve for the time interval k, <L(k)>, as the average value over k sets of L_m(k). If <L(k)> α k ^–D, then the curve is fractal with the dimension D.

Fractal dimension for daily sunspot numbers:

We make a new time series with daily sunspot numbers for a year and calculate <L(k)> as defined above. Then we plot the logarithm of length, log <L(k)>, as a function of log k. The unit of k is a day in this case. If <L(k)> α k^–D, we judge that the curve is fractal. Then we deduce fractal dimension from the slope of a plot.

Table 1. An example of calculated values of <L(k)> for the daily sunspot number, 2001

K	*< L(k )>*
1	*14468*
2	*3656*
3	*1580.889*
4	*1015*
5	*621.32*
6	*410.1389*
7	*326.5102*
8	*251.7031*
9	*174.8395*
10	*148.74*
11	*114.4628*
12	*45.71528*
13	*90.30177*
14	*78.84184*
15	*67.3777*
16	*64.90625*
17	*57.6609*
18	*48.81173*
19	*46.32687*
20	*41.7725*
21	*33.3673*
22	*31.3657*
23	*29.2949*
24	*16.61806*
25	*25.936*
26	*24.3580*
27	*21.8395*
28	*21.958*
29	*21.03568*
30	*18.86652*
31	*18.40362*
32	*17.13562*
33	*14.27078*
34	*14.15906*
35	*13.33867*

An example of the curve length <L(k)> for the time series of daily sunspot numbers in 2001, on a doubly logarithmic scale was plotted and is shown in Figure 1. We can determine fractal dimension from the slope of this plot.

In the case of 2001 data, the curve breaks at about 10 days. The fractal dimension is 1.47 within 10 days and is 1.98 for longer than 10 days. A time series is originally one-dimensional data; hence fractal dimension is 1 for a regular time series, and it is 2 for a completely random time series. Fractal dimension for the shorter time scale expresses more regular variation than one for the longer time scale shown in Figure 1. Fractal dimension, 1.98, for the longer time scale is fairly close 2. This implies randomness of the time series and it is difficult to predict this time range.

Annual variation of the fractal dimension for sunspot number:

Yearly variation of fractal dimension, deduced by daily sunspot numbers, is shown in Figure 2. Yearly sunspot numbers vary with the 11-year solar cycle. However, yearly values of fractal dimension do not change in correspondence to solar activity.

CONCLUSION:

The fractal dimension method was adopted to examine a time series of daily sunspot numbers. We can evaluate randomness of a time series by determining fractal dimension. The average fractal dimension between 1994 and 2008 was about 1.49 within 10 days and about 1.92 for periods longer than 10 days. This result reveals the possibility of short term prediction and the difficulty of medium term prediction (longer than 10 days). Sunspot numbers vary according to the 11-year solar cycle. However, annual values of fractal dimension of <L(k)> do not change in concert with this cycle. This may suggest that the physical mechanism producing short and medium scale time variation does not change throughout the 11-year cycle.

REFERENCES

1. G.P.Pavlos, D.Dialetis, G.A. Kyriakou & E.T.Saris., A Preliminary low-dimensional chaotic analysis of the Solar cycle, Annales Geophysicae ,1992, 759-762, 10

2. O.Chumak., Self-similar and self-affine structures in the observational data on solar activity, Astronomical and Astrophysical Transactions, 2005, 93-99, 24

3. Gorney,D.J., Solar Cycle Effects on the Near-Earth Space Environment, Reviews of Geophysics, 1990, 315, 28

4. Higuchi,T., Approach to an irregular time series on the basis of the fractal theory, Physica D, 1988, 277, 31

5. Koons,H.C. and Gorney,D.J., A Sunspot Maximum Prediction Using a Neural Network, EOS, May 1, 1990.

Received on 25.01.2011

Accepted on 05.02.2011

Research J. Science and Tech. 3(1): Jan.-Feb. 2011: 25-27