Wavelet Analysis

Figure 6.3: Wavelet power transform of a typical network light curve.

Similar to Chapter 5 a wavelet analysis was carried out on each light curve using a Morlet wavelet of the form,

For the Morlet wavelet, the Fourier period is ~1.03 times the wavelet scale (Torrence & Compo 1998). A typical wavelet power transform (square of the absolute value of the wavelet transform) for a network light curve is shown in Figure 6.3. In Chapter 5 the wavelet power transform is plotted as frequency against time (see Figures 5.5 and 5.6). However for the work in this present chapter it more apt to plot period against time. Hence in Figure 6.3 the abscissa is time, t, and the ordinate is period, P. This is displayed on a linear intensity scale, such that brighter areas correspond to greater oscillatory power. The contours are at the 95% significance level and the two slanted lines define the cone-of-influence (COI). Edge effects are significant above these two lines. The extent of the COI, from the beginning and the end of the time series, at each period P, is defined in Torrence & Compo (1998) for the Morlet wavelet, as the decorrelation time for a spike in the time series, t_d =  (√2)P/1.03 s . For example, for the dashed lines on Figure 6.3 at P = 1200 s, the COI extends from t = 0-1647 s, and from t = 5001-6648 s. After creating the wavelet power transform, an automated routine calculated the lifetime and periodicity of any oscillations following the procedure first described in Ireland et al. (1999). First, all power below the 95% significance level or above the COI lines was removed. The routine then searches for any remaining power maxima. The lifetime of the oscillation at the period of each power maximum is defined as the interval of time from when the power reaches above 95% significance to when it dips below 95% significance again. The lifetime was then divided by the period to give a lifetime in terms of complete cycles. For example, for the oscillatory power in contour A in Figure 6.3, the lifetime is given by (4900-3500)/300 = 4.67 cycles. Any oscillations lasting for less than t_d (e.g., for the maxima in contour C in Figure 6.3, lifetime = (5100-4400)/650 = 1.08 cycles) were discarded as possibly being due to a spike in the time series. From the criterion of a lifetime of at least (√2)P/1.03 outside the COI (of size 2 (√2)P/1.03, the maximum possible detectable period, P_max, can be obtained from,

where δt is the cadence, and N is the number of data points in the light curve (hence δt x (N-1) is the total duration of the light curve). Rearranging for the dataset used here,

In Figure 6.3 P_max is shown as the dash-dot line, and is the value used as the cut-off for the high pass filter (Section 6.3.3). The final output from the routine is a list of lifetimes and periodicities for all oscillations in each input light curve.