In a first attempt to analyze the burst, quiet Sun, and sky time profiles, sliding power spectra were used, but this did not reveal clear-cut patterns. The power spectra calculated from different time intervals have quite different shapes, i.e., they depend on time essentially. Therefore, we check whether the SFA and the MRA are more convenient to describe these non-stationary burst time profiles (Kurths et al. 1995). It is important to note that no filtering is required to apply these methods.
Figure 5: Double logarithmic plot of the structure function estimated (diamonds) from the radio flux of the event presented in Fig. 1 (click here). Thin solid lines show the linear fits used to derive the scaling exponents
The concept of self-affinity presented in the following is a
promising approach to describe a uniform broad-range scaling in time,
although the underlying process can be non-stationary (Feder
1988). This concept is a generalization of self-similarity, which is
the basis of (deterministic) fractal geometry.
A convenient way to quantify self-affinity is based on the structure function (SF; two-point
where is the average over time t, and takes integer values. denotes the sampling time and x(t) is the observed time series. If a process is self-affine then this SF obeys the following power-law scaling with
where H is the characteristic scaling exponent. Self-affinity means that a scaling exponent H does exist independently of the shift in time. A typical example of a self-affine process is the fractional Brownian motion (0 < H < 1) which generalizes the classical Brownian motion or random walk, where H = 0.5 (Feder 1988). Such a process was investigated by Osborne & Provenzale (1989). Starting from a power-law decay of its power spectrum ,
with spectral index , they construct a stochastic time series by
where (k = 1, ..., N/2), with , and . The were chosen at random. Such fBm is characterized by a power-law scaling in time (Eq. 2 (click here)) as well as in frequency
Both scaling exponents are simply related by
It is important to note that the fBm process is not stationary because the standard deviation of the time series depends on the length of the time interval in which it is calculated. This standard deviation also scales as a power-law
Therefore, fBm cannot be characterized by the correlation dimension. The main difference to self-similar processes is that fBm exhibits a different scaling in space and time. The concept of fractal dimensions requires, however, that both scalings are identical.
For H = 0.5 in Eq. (2 (click here)), we have the classical Brownian motion, i.e. the increments are not correlated. In the case H > 0.5, there is a positive persistence. Therefore, an increasing trend in the past implies an increasing trend in the future, i.e. processes with H > 0.5 are characterized by long-range correlations.
Since the analyzed microwave bursts are far from being stationary, we calculate the SF, (Eq. 1 (click here)), of the radio flux time profiles. Surprisingly, for each event we find a well-expressed power-law scaling (cf. Fig. 5 (click here) and Table 1 (click here)) in a rather broad region of . It turns out that two of the analyzed events are compatible with the usual Brownian motion. The other ones are characterized by scaling exponents significantly larger than 0.5 which is typical for fBm with rather long-range correlations (Table 1 (click here)).
If Eq. (2 (click here)) is applied to white noise, H=0 is obtained. In fact, the quiet Sun and sky observations yield H which approach that value (Table 1 (click here)).
The power-law scaling of the SF of the radio flux as well as zooming of the time profile suggest that there is no dominating narrow-band time scale. It is important to note that the SF, like the power spectrum, yields only global properties of the temporal behavior of the radio flux. In the following we present the MRA which is more suitable to quantify local and time-dependent phenomena, which are typical of solar radio bursts.
In the general case of non-stationarity, we have to expect an inhomogeneous scaling behavior, varying in space and/or in time. Detailed information on both, the location and the size of the characteristic features is of interest here. Wavelets are a proper tool to analyze such phenomena. It should be mentioned that the well-known windowed Fourier transform is another tool to study local behavior, but it is a much more coarse-grained one (cf. Daubechies 1992; Scargle 1993; Scargle et al. 1993).
We first recall a few basics about the most common global technique, the power spectrum, which is based on the Fourier transform (Priestley 1981). It is an efficient tool for giving some dominant frequencies (or characteristic sizes). This transform is a projection on an orthogonal basis consisting of harmonic functions. Hence, there exists a unique decomposition and reconstruction formula for a given function x(t), but there is no simple relationship between the local behavior of x(t) and the Fourier coefficients. This information is so deeply buried in the phases of the coefficients that it is very difficult to retrieve.
In generalization, the notion of wavelet analysis addresses both,
unknown periodicities and non-stationary structures.
The wavelet analysis is based on time-limited elements, the wavelets.
By this means, one has the possibility of dealing with
non-stationary time series, where, e.g., some coherent structures
evolve in time.
The wavelet transform of x(t) is the decomposition into a basis of
functions wa,b(t) with
all derived from a unique function w(t), called the "mother wavelet,'' by translation b and scaling a. Several functions have been recommended as wavelets, e.g., Daubechies wavelets and Gabor-Malvar wavelets. For our purpose, the simple triangle-like wavelets (cf. Mallat 1989; Meyer & Ryan 1993; Vigouroux & Delache 1994) are appropriate, because their shape fits the time profile of the radio flux quite well.
We use a bi-orthogonal wavelet basis consisting of the
collection wj,k(t), where is the scaling factor and
(set of relative integers) is the translation,
together with q(t-k), , where q(t) is
a smooth function with a rapid decay.
On this basis, x(t) can uniquely be written as
The two functions q(t) and w(t) cannot be chosen independently. Among the many possibilities for the two functions q(t) and w(t), we choose, following Bendjoya et al. (1993),
A main advantage of wavelets is the possibility to calculate the coefficients ckj and dkj recursively. The first trivial step is to take the function q(t) with the same resolution as the sampled signal. We only have to compute the ck0 (the superscript refers to the number of the step in the iterative process). From the formula for q(t), it is easy to see that the ck0 are simply the sampled values of x(t).
Then we scale the resolution. We choose the normalizing coefficients such that we replace q(t) by q(t/2)/2. There is only one level of fine fluctuations to add to the smoothed part to recover the signal, and it is defined by the set of d1,k = dk1. We can now repeat the same procedure to the smoothed part of level 1. This gives two new sets of coefficients ck2 and dk2.
With our choice of q(t) and w(t), the formulae to
obtain the ckj and the dkj are simple recursions (Mallat
These equations can easily be interpreted in terms of filtering. The ckj are obtained by applying a low-pass filter to the ckj-1 (Eq. 12 (click here)). On the other hand, the dkj being obtained by a difference between two levels of ckj are in fact the result of a band-pass filter applied to the signal (Eq. 13 (click here)). This algorithmic scheme is called time-scale analysis or multiresolution analysis (MRA).
The decomposition described above is best portrayed in a - - plot of the coefficients dkj, called scalogram (Figs. 6 (click here)-9 (click here)), which shows the scaling behavior of the radio flux x(t) in dependence on the time location . Different intensities of the coefficients (amplitudes) are displayed by different grey levels. The computational effort of the chosen recursive procedure is similar to that for calculating the fast Fourier transform.
To illustrate the MRA, we take the following
function (Fig. 10 (click here))
and distorted it somewhat by adding noise. Such a time series is non-stationary, but its characteristics are displayed quite well by the scalogram (Fig. 10 (click here)).
Figure 6: Scalogram of the event presented in Fig. 1 (click here). We have plotted the normalized positive wavelet coefficients. White pattern represent dominant time scales. The dark background represents wavelet coefficients with very small amplitudes. For clarity, 5 sec averages are plotted
Figure 7: Scalogram of the event presented in Fig. 2 (click here)
Figure 8: Scalogram of the event presented in Fig. 3 (click here)
Figure 9: Scalogram of a fBm realization with presented in Fig. 4 (click here)
Figure 10: Illustration of the MRA. Left: Wake-like data according Eq. (14 (click here)
). Right: Scalogram. The scalogram shows clearly both the location and the width of the dominant features of the data