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4. Scalograms and scalegrams of radio fluxes


Figure 11: Scalegrams for different time intervals of the event presented in Fig. 1 (click here). Pre-burst phase (dotted line; squares): tex2html_wrap_inline1885, main phase (solid line; diamonds): tex2html_wrap_inline1887, post-burst phase (dashed line; triangles): tex2html_wrap_inline1889

Figure 12: Scalegrams of the event presented in Fig. 2 (click here). Dotted line (squares): tex2html_wrap_inline1891, solid line (diamonds): tex2html_wrap_inline1893, dashed line (triangles): tex2html_wrap_inline1895

Figure 13: Scalegrams of the event presented in Fig. 3 (click here). Dotted line (squares): tex2html_wrap_inline1897, solid line (diamonds): tex2html_wrap_inline1899, dashed line (triangles): tex2html_wrap_inline1901

Figure 14: Scalegrams of a fBm realization with tex2html_wrap_inline1871 presented in Fig. 4 (click here). The three different scalegrams are calculated for the same intervals as in Fig. 13 (click here)

Figure 15: Linear fits to the slopes in the power law regions of the scalegrams presented in Fig. 11 (click here). Error bars associated with these fits are shown and also given in the last column of Table 1 (click here)

Figure 16: Same as above for Fig. 12 (click here);

Figure 17: Same as above for Fig. 13 (click here)

Figure 18: Same as above for the fBm realization presented in Fig. 14 (click here)

Next, we apply the MRA to the microwave burst events listed in Table 1. The problem we are interested in is a description of a broad band of time scales in microwave bursts, in order to support the diagnostics of the underlying coronal energy release processes (Krüger et al.  1994).

Using the functions given by Eqs. (10 (click here)) and (11 (click here)), we observe that the positive wavelet coefficients dkj reflect the burst-like behavior of the radio flux quite well. In Figs. 6 (click here)-9 (click here) the logarithm of the normalized positive wavelet coefficients tex2html_wrap_inline1907 is plotted (scalogram). Summing up the wavelet coefficients over time (index k), we obtain a picture similar to a power spectrum, the so-called scalegram (Figs. 11 (click here)-14 (click here)):
This allows to calculate the spectral index tex2html_wrap_inline1601 from the slope of that part of the scalegram which follows a power law (Flandrin 1994). Figure 14 (click here) shows that, for a process with structural similarity to fBm, the lengths of the available time series permit to derive the scaling up to about 16 s. The observations in fact show a scaling (power law behavior of the scalegram) from the limit set by the resolution (1 s) up to tex2html_wrap_inline1913 (Figs. 11 (click here)-13 (click here)) with minor deviations for the shortest data sets (<500 data points). For this range the spectral indices tex2html_wrap_inline1601 have been estimated; they are shown in Figs. 15 (click here)-18 (click here) and listed in the last column of Table 1 (click here). They agree quite well with the spectral indices calculated from the structure function.

The wavelet transform calculated from the data indicates that a rather broad range of time scales from 1 s to a few minutes is involved during most parts of the events (Figs. 6 (click here)-8 (click here), 11 (click here)-13 (click here)). The sequences of bright patterns in Figs. 6 (click here)-8 (click here) give an impression of the dominant scales at different times during the bursts. Indications for hierarchic time structures (where different, well separated time scales become dominant) are only very weak and short-lived. For example, there are short periods around tex2html_wrap_inline1919 and tex2html_wrap_inline1921 in Fig. 6 (click here) and around tex2html_wrap_inline1923 in Fig. 7 (click here) where two maxima of the wavelet coefficients do exist.

For purposes of comparison we have also calculated the scalograms for several kinds of surrogate data, such as white noise, linear colored noise (autoregressive processes), and fBm, using comparable data lengths. As expected, for white noise the scalogram exhibits complete disorder, whereas periodic features are transformed into a simple vertical stripe pattern. These stripes are located at the maxima of the periodic signal. The vertical length of the stripes corresponds to the half width of the period length (see also Fig. 10 (click here)).

If we compare the scalogram plots of these different models with the data, we find that the scalogram of an fBm with H = 0.625 (Fig. 9 (click here)) looks very similar to that of the bursts. This agreement is strengthened by the spectral index tex2html_wrap_inline1601 calculated from the slope of the scalegram, which nearly equals tex2html_wrap_inline1929.

It is important to note that the scalograms from an off-Sun position and from a quiet region on the Sun differ significantly from those of the bursts. The corresponding spectral indices, given in Table 1 (click here), show that long-range correlations are missing in these time series (H<0.5).

Next, we analyze the different phases of the burst separately, i.e. the spectral index tex2html_wrap_inline1601 is derived for the pre-impulsive phase, the main phase, and post-impulsive phase of the bursts shown in Figs. 1 (click here)-3 (click here) separately. Surprisingly, we find that the spectral indices tex2html_wrap_inline1601 change only slightly between these intervals. Thus the three phases appear to be structurally analogous, i.e., the relative contribution of the different time scales remains roughly constant. There is a tendency for tex2html_wrap_inline1753 to be slightly smaller in the post-impulsive phase, indicating that the emission becomes more random (less correlated) in that phase (Figs. 15 (click here)-17 (click here)).

In order to check the reliability of the scalegrams, we have calculated them in each case for 40 slightly different time intervals of equal length. We observe in all cases that the used method is rather robust. The error bars obtained from these sets of 40 scalegrams for each analyzed time interval are comparable to the symbol size in Figs. 11 (click here)-14 (click here).

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