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Astron. Astrophys. 328, 670-681 (1997)

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1. Introduction

Solar activity is a quasi-periodic phenomenon, with a main period of approximately 22 years. This periodicity may not be constant, though. One way of studying the long-term behavior of this cycle is to use the sunspot number as an index of the 11y, or Schwabe, cycle. Wolf (1851) started such a reconstruction last century, providing the so-called Wolf number. However, Wolf did not look up all the existing archives and used different indices (such as the magnetic aurora) to fill in gaps in sunspot observations. Recently, Hoyt and Schatten (1992 a,b) have undertaken a complete re-analysis of the archives, using sunspot groups to reconstruct solar cycles. As a result, a fairly homogeneous sample of monthly sunspot groups is available spanning a 385-year period (Hoyt et al. 1994). The long-term sunspot forecast started at the time of Galileo and became systematic under the French Astronomical School (Picard and La Hire), from 1660 to 1719. From the 18th century onward, this observational program has been pursued by an increasing number of observatories. This long sunspot survey gives us the possibility to analyze solar activity periodicities in detail, in particular, to find out how stable the main period is, what additional periodicities are present, and what connection there may be between solar periodicities and the Maunder minimum.

The simplest technique for investigating periodicities in sunspot data is obviously the Fourier analysis, i.e. the comparison between the record and a sinusoidal signal of a given frequency. However, this method is not best appropriate to correctly interpret the data. The main (11-year) signal is pseudo-periodic, and its amplitude changes by several orders of magnitude over the time interval under consideration. There is another pseudo-periodic signal, referred to as the Gleissberg cycle, the duration of which ([FORMULA] 100 years) is only a few times less than the time interval spanned by the whole time-series (four centuries). Furthermore, a straightforward application of the Fourier analysis to such data set may lead to spurious periodicities, especially at low and intermediate frequencies, due to the finite size of the time-series (Ribes et al., 1989).

[FIGURE] Fig. 1. Examples of wavelets: a Mexican hat; b Morlet wavelet.

A few methods have been developed for studying local spectral properties of quasi-periodic signals (time-frequency analysis). The Wigner-Ville algorithm, for example, produces a spectral analysis of an autocorrelation function (Altarac, 1995), and the windowed Fourier transform of Gabor is based on a family of harmonic functions with different frequencies but limited by the constant size of the window. The wavelet technique is more appropriate to the study of quasi-periodic signals. The wavelet transform, in contrast to the Wigner-Ville technique, yields a linear representation of the signal, and in contrast to the Gabor transform, it uses a family of self-similar functions (for comparison and discussion, see Toresanni, 1995). The wavelet analysis is more appropriate than the Fourier technique for denoising the signal (see Donoho, 1994); this is appreciable when time-series (like ours) contain observations of uneven quality.

The aim of our paper is to apply wavelet analysis to monthly sunspot group data and discuss the physical results obtained by this technique. Such an analysis was already performed by Ochadlick et al. (1993) using the yearly sunspot data starting from the year 1700. Our study differs from this previous one in that: 1-) Our time-series starts in the early 1600's, including the remarkable event known as the Maunder minimum. 2-) We use monthly sunspot groups rather than yearly means, which seem to be too crude when the averaging time is only 11 times less then the main sunspot period. Because of the time interval spanned, Ochadlick et al. (1993) could hardly make out the Gleissberg cycle. 3-) We also made use of the algorithms developed by Torresani (1995), and applied to the wavelet technique to search for frequency modulation laws in the signal.

Recently, Lawrence et al. (1995) examined daily sunspot data over the last two decades. They obtained the fractal properties of solar activity indicating chaotic behavior at timescales of the order of 1 year, which is much less then the basic period of solar activity. They found that the chaotic behaviour seems to be connected mainly with sunspot formation rather than with solar activity itself. We are interested in longer timescales, so our analysis of monthly sunspot data over four centuries is complementary to that of Lawrence et al. (1995).

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© European Southern Observatory (ESO) 1997

Online publication: March 26, 1998