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Astron. Astrophys. 322, 835-840 (1997)

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3. Data analysis

Several methods are generally used to study the time-dependence of a periodic signal. A very popular example from variable star research is the so-called `O-C diagram' in which deviations between the observed and the predicted zero-phases of the signal are depicted over time. This method requires a sufficient signal-to-noise ratio, and good phase coverage because the signal must be `visible'. Another often-used method is to sub-divide the time series, and analyzing each sub-sample separately through periodogram analysis (Scargle 1982; Horne & Baliunas 1986). Using this method, Baliunas et al. (1985) and Donahue (1993) examined the year-to-year variations of the rotation periods of several solar-like stars (stellar butterfly diagrams). A more sophisticated method is the collection of Fourier transforms of subsamples formed by a window propagating over the entire series. This constitutes the Gabor transform. However, one disadvantage of the Gabor transform is that the number of oscillations inside the window depends on the frequency analyzed. Thus, the relative frequency resolution will vary over the frequency interval studied.

The wavelet transform uses a window whose width is a function of the actual frequency. Several types of wavelets can be used. However, if the signal is sinusoidal, the wavelet should also be chosen to be sinusoidal. Thus, the Morlet wavelet (Grossmann & Morlet 1984)

[EQUATION]

represents a sinusoidal oscillation contained within a Gaussian envelope. Then the wavelet transform can be written as

[EQUATION]

where [FORMULA] is the complex conjugate of W. The discrete wavelet transform is

[EQUATION]

where [FORMULA]. The parameter a controls the resolution in both frequency and time (Baudin et al. 1994):

[EQUATION]

Therefore, the relative frequency resolution is uniquely determined by a, whereas the time resolution depends on the frequency itself to hold the number of oscillations inside the wavelet constant. We choose [FORMULA] in our analysis in order to balance the time and frequency resolution.

To evaluate the time-frequency behaviour we plot a wavelet map as the square of WT([FORMULA]) (the `power') with dependence on time and frequency as a first step. A visual inspection of such a plot shows regions of enhanced power which, in case of a positive detection of the SDR, will follow systematic trends in the form of one or more `mountain ranges' or, even in a more complicated manner, as a series of isolated peaks as the peak frequency changes over the course of the activity cycle. Next, cuts are made along the frequency axis at discrete values of time, in order to find the absolute maximum power and also the relative maxima of power at each epoch and their corresponding frequencies. In the third step, these frequencies are plotted over time. This plot will represent a stellar butterfly diagram, i.e. the time evolution of rotation periods that are defined by the rotation experienced by plage regions causing modulation of the emission core of the solar disk-integrated Ca II K line.

As in the case of Fourier analysis of unevenly sampled data, artifacts may result which can lead to misinterpretation of the wavelet map. Szatmáry, Vinkó & Gál (1994: SVG94) investigated the properties of the wavelet map in the cases of the following signal characteristics: i) varying amplitude, period, and zero-phase; ii) strong deviations of the signal from a sine curve; iii) double-mode oscillations; iv) the influence of noise; and, v) gaps in the time series. Based on their analysis, we assume the following expected characteristics might be found in wavelet pictures of our time series:

i) Fig. 2 shows that strong variations in amplitude, period, and phase exist, even on timescales close to the rotation period. A variable amplitude will lead to a variation of power with time, whereas jumps in zero-phase can disrupt a `mountain range' into isolated peaks. A further loss of power occurs because the original data are unevenly sampled, and while filling in the time series with zeroes causes it to appear evenly spaced, the additional points complicate the wavelet map by redistributing power randomly. A similar effect results from the existence of seasonal gaps in the data that are typical for stellar observations, although not present in the solar time series considered here. While the systematic changing of the period by SDR should be visible in the wavelet map the Fourier analysis of the time series as a whole will lead to a broadening of the Fourier peak only, making it impossible to derive period shifts and jumps within the time series. The ability to investigate the time-dependence of the oscillating parameters is an important advantage of wavelet analysis. Furthermore, jumps of the phase caused by the eruption of multiple ARs at different stellar longitudes should be visible in the wavelet map, while this same behaviour causes frequency splitting in the discrete Fourier transform (Donahue 1993). However, SVG94 uses a sufficiently short window in order to resolve the temporal behaviour in detail. In our case, the time-resolution will be chosen to be low enough to hold the frequency resolution required. In fact, in selecting [FORMULA] the corresponding time resolution ([FORMULA]  d) will likely smooth much of the variability at shorter time-scales as might be caused by ARGD. Hence, the evolution of ARs might remain invisible. One can expect, therefore, that zero-phase modulation as caused by ARGD at different stellar longitudes will produce some artifact in the form of side-lobes in the wavelet map. The reason is quite simple: as the time-resolution decreases, the differences between Fourier analysis and wavelet analysis also decrease. Fig. 12 of SVG94 shows that zero-phase modulation will produce a complex pattern of peaks in the wavelet map at high time-resolution whereas the Fourier spectrum shows side-lobes. We can therefore assume that the corresponding wavelet map of low time-resolution will also show side-lobes in form of mountain ranges beside the dominant mountain range.

ii) Deviations from a sine curve will have little influence because strong deviations such as those in the light curve of eclipsing binaries cannot be expected.

iii) Double-mode oscillations can arise from two ARs at different latitudes. The behaviour of the wavelet spectrum will depend on the time resolution because the superposition of two signals leads to a complicated but regular pattern in the time series. If the time-resolution is fine enough, the pattern will be visible in the wavelet map; otherwise the wavelet picture is no more informative than the Fourier spectrum, i.e., there will be two parallel mountain ranges in the wavelet map. If the two modes are close together in frequency; a beat phenomenon is expected. In this case, the wavelet map looks like amplitude modulation. Mode switching can, in principle, be localized in time which cannot be examined using Fourier analysis.

iv) The influence of low amplitude noise on wavelet analysis is similar to its effect on Fourier analysis. However, in the case of strong noise, the Fourier spectrum is a better tool simply because the maximum number of data points is included in a window that has the maximum length possible.

v) Gaps in the data are quite usual for stellar observations. Under certain circumstances they can produce side-lobes in the wavelet map, similar to `aliases' seen in Fourier analysis (Horne & Baliunas 1986). In this particular investigation, large (seasonal) gaps are not present; the characteristic gap size in this time series is only on the order of a few days.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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