Figure 11: Scalegrams for different time intervals
of the event presented in Fig. 1 (click here).
Pre-burst phase (dotted line; squares): 
, main phase (solid
line; diamonds): 
, post-burst phase (dashed line;
triangles): 
Figure 12: Scalegrams of the event presented in
Fig. 2 (click here).
Dotted line (squares): 
, solid line (diamonds): 
, dashed line (triangles): 
Figure 13: Scalegrams of the event presented in
Fig. 3 (click here).
Dotted line (squares): 
, solid line (diamonds): 
, dashed line (triangles): 
Figure 14: Scalegrams of a fBm realization with 
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 
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 
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 
(Figs. 11 (click here)-13 (click here)) with minor
deviations for the shortest data sets (<500 data points).
For this range the spectral indices 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 
and 
in
Fig. 6 (click here) and
around 
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
calculated from the slope of the scalegram, which nearly equals
.
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 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 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 
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).