Multiresolution analysis of solar mm–wave bursts

Astronomy and Astrophysics Supplement Series, Jul 2018

Two methods of multi-scale time series analysis are applied to solar mm-wavelength flux time profiles in order to assess the diagnostic power of these tools for the exploration of nonlinear energy release processes. Both the multiresolution analysis (MRA), a method based on the wavelet transform, and the structure function analysis (SFA) permit the treatment of non-stationary time series. In addition, the MRA offers a local decomposition of the scaling behavior of the flux variations. Our main emphasis is directed at a decomposition of the contributions of the different time scales to the overall flux profile. The methods yield consistent values of the "spectral index

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Multiresolution analysis of solar mm–wave bursts

ICARUS Multiresolution analysis of solar mm{wave bursts U. Schwarz - 2 J. Kurths 2 B. Kliem 1 A. Kru¨ger 1 S. Urpo 0 0 Helsinki University of Technology , Otakaari 5a, SF-02150 Espoo , Finland 1 Astrophysical Institute Potsdam , An der Sternwarte 16, D-14482 Potsdam , Germany 2 Institut fu ̈r Theoretische Physik und Astrophysik, Universita ̈t Potsdam , Am Neuen Palais, D-14469 Potsdam , Germany Two methods of multi-scale time series analysis are applied to solar mm-wavelength flux time pro les in order to assess the diagnostic power of these tools for the exploration of nonlinear energy release processes. Both the multiresolution analysis (MRA), a method based on the wavelet transform, and the structure function analysis (SFA) permit the treatment of non-stationary time series. In addition, the MRA o ers a local decomposition of the scaling behavior of the flux variations. Our main emphasis is directed at a decomposition of the contributions of the di erent time scales to the overall flux pro le. The methods yield consistent values of the spectral index" which describes the scalings contained in the time series. We nd that time pro les of bursts are qualitatively analogous to fractional Brownian motion (fBm), possessing long-range temporal correlations. Such correlations are not found in quiet Sun observations. The MRA of six solar mm-wave bursts shows that the radio flux is always composed of contributions from a broad range of time scales. Also during the main phase of bursts, which appears to be structurally analogous to the pre- and post-burst phases at a resolution limit of 1 s, flux fluctuations are enhanced in a broad range of time scales. This suggests that the mm-wave bursts are composed of unresolved elements, just as the pre- and post-burst time pro les. The underlying energy release thus appears to be fragmentary. These results are discussed in terms of the avalanche model and plasma physical models for solar energy release events. Sun; radio radiation | Sun; flares | methods; data analysis 1. Introduction Solar radio bursts are typical examples of superpositions of transient phenomena, i.e. the underlying physical processes are non-stationary (Isliker & Kurths 1993). To study the energization and emission processes that cause bursts, data analysis methods are required which can be applied to such non-stationary time series. Promising methods of this kind are the structure function analysis (SFA) and the multiresolution analysis (MRA), whose potential in the investigation of such bursts is discussed here. In particular, the wavelet transform of the MRA permits a local decomposition of the scaling behavior in a time series { in contrast to methods designed to detect global properties, such as the Fourier analysis and the SFA. Using these methods we look for structural di erences between time series of the flux of di erent sources (bursts, quiet Sun, and sky background) as well as between di erent phases of the bursts at mm-wavelengths. Statistical studies of solar energy release events, e.g. the distribution of event number versus energy content as observed at hard X-rays, have led to a description in terms of avalanches in a corona which has stored energy and is in a state of self-organized criticality (Lu & Hamilton 1991). The observed power-law distribution naturally follows from that model. One basic property of this model is that the system under consideration has no characteristic spatial scale above an elementary scale of the smallest avalanche (the smallest energy release event), up to the system size, the size of active regions. This elementary scale, which is a characteristic of the involved plasmaphysical process, is below the current resolution limit, since the power-law distribution extends down to the resolution limit. The success of this approach suggests that the energy release is basically fragmentary, the events being composed of elementary building blocks. Many attempts have been made to resolve the elementary building blocks of the energy release and possibly also larger characteristic scales from the time pro le of the solar flux, particularly in the radio and the hard X-ray ranges. Most of these studies have concentrated on the shortest time scales detectable. For example, Kiplinger et al. (1983) have found fast hard X-ray spikes with duration down to 45 msec and Gu¨del & Benz (1990) have found similar durations in decimeter radio wavelength spike burst observations. Here we follow a di erent approach by attempting to resolve a broad range of time scales above 0.5 s present in a flux record of the emission at millimeter wavelengths. Thus, we quantify the general structure of the emission and study its evolution in time, and we are able to investigate whether a hierarchy of dominant time scales is present in the flux pro le, as has been discussed previously (Kru¨ger et al. 1987, 1994), or whether the distribution of contributing time scales is more or less structureless, as predicted by the avalanche model. In comparison to hard X-ray data, the choice of this wavelength range has the advantage that the flux prole before and after bursts can also be analyzed and compared with the burst data (since detector or sky noise contributions remain small), and in comparison to the decimetric range the mm-wavelength flux pro le is supposed to reflect the energy release process in a more direct manner, less influenced by nonlinear plasma processes which may modulate the radio wave intensity. The organization of this paper is as follows: In the following section we present the data. In Sect. 3 we introduce the tools of data analysis: SFA and MRA. The diagnostic capabilities of these tools in the application to the radio burst time pro les are discussed in Sect. 4. In Sect. 5, we discuss the physical interpretation of some features of the obtained results, and Sect. 6 presents the conclusions. 2. The microwave data We analyze several microwave bursts digitally observed at 36.8 GHz (8.2 mm wavelength) at the Mets¨ahovi Radio Research Station of the University of Technology Helsinki/Finland (cf. Table 1). The receiver is connected those diverse flux pro les can be quantitatively described with a Cassegrain telescope of 14 m diameter resulting in in a uniform manner. a half-power beam width of 2.4 arcmin at this frequency. For later comparison, we add here in Fig. 4 the time The quiet-Sun level at this frequency is 7800 K (Urpo pro le of a synthetic realization of a fractional Brownian et al. 1992). Typical time pro les are shown in Figs. 1-3. motion (fBm) process with a spectral index similar to that Each data set has a duration of about one hour with a of the radio flux pro le shown in Fig. 3. time resolution of 0.5 s. The analyzed bursts originated in AR 6555 (Solar Geophysical Data). The observations are chosen because 3. Data analysis they show variations in a rather broad range of time scales. In particular the events of Figs. 1 and 2 exhibit short- In a rst attempt to analyze the burst, quiet Sun, and sky period radio spikes which are known to appear frequently time pro les, sliding power spectra were used, but this at dm-waves but are rare at solar bursts in the short did not reveal clear-cut patterns. The power spectra calcm- and mm-wave ranges (Slottje 1978; Benz 1986; Benz culated from di erent time intervals have quite di erent et al. 1992). The event shown in Fig. 3 appears to possess shapes, i.e., they depend on time essentially. Therefore, a di erent temporal dynamics than the other two bursts. we check whether the SFA and the MRA are more conveMoreover, the temporal structure of the pre-burst flux pro- nient to describe these non-stationary burst time pro les le in Fig. 1 appears to di er from that of the post-burst (Kurths et al. 1995). It is important to note that no lterpro le. This raises the question whether the structure of ing is required to apply these methods. The concept of self-a nity 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-a nity is based on the structure function (SF; two-point correlation function) S( ) = hjx (t + t) − x (t) j2it where h:::it is the average over time t, and takes integer values. t denotes the sampling time and x(t) is the observed time series. If a process is self-a ne then this SF obeys the following power-law scaling with S( ) = where H is the characteristic scaling exponent. Selfa nity means that a scaling exponent H does exist independently of the shift in time. A typical example of a self-a ne 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 P (!k), P (!k) = C !k− ; with spectral index series by (3) , they construct a stochastic time that fBm exhibits a di erent scaling in space and time. The concept of fractal dimensions requires, however, that both scalings are identical. For H = 0.5 in Eq. (2), 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, S( ) (Eq. 1), of the radio flux time pro les. Surprisingly, for each event we nd a well-expressed power-law scaling (cf. Fig. 5 and Table 1) 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 signi cantly larger than 0.5 which is typical for fBm with rather long-range correlations (Table 1). If Eq. (2) 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). The power-law scaling of the SF of the radio flux as well as zooming of the time pro le 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. 3.2. Multiresolution analysis N=2 In the general case of non-stationarity, we have to exxi = X k cos (!ki t + 'k) ; i = 1; :::; N; (4) apnecdt/oarn iinnhtoimmeo.geDneetoauilsedscai nlifnogrmbeahtiaovnioorn,vbaorythin,gthine slpoaccaek=1 tion and the size of the characteristic features is of inwhere !k = k ! (k = 1; :::; N=2), with ! = 2 =N t, terest here. Wavelets are a proper tool to analyze such and k = pP (!k) !. The 'k 2 [0; 2 ) were chosen at phenomena. It should be mentioned that the well-known random. Such fBm is characterized by a power-law scaling windowed Fourier transform is another tool to study local in time (Eq. 2) as well as in frequency behavior, but it is a much more coarse-grained one (cf. Daubechies 1992; Scargle 1993; Scargle et al. 1993). P (f ) / f − : (5) We rst recall a few basics about the most common Both scaling exponents are simply related by global technique, the power spectrum, which is based on the Fourier transform (Priestley 1981). It is an e cient − 1 tool for giving some dominant frequencies (or characterisH = 2 : (6) tic sizes). This transform is a projection on an orthogonal basis consisting of harmonic functions. Hence, there exists It is important to note that the fBm process is not station- a unique decomposition and reconstruction formula for a ary because the standard deviation X of the time series given function x(t), but there is no simple relationship depends on the length of the time interval in which it is between the local behavior of x(t) and the Fourier coe calculated. This standard deviation also scales as a power- cients. This information is so deeply buried in the phases law of the coe cients that it is very di cult to retrieve. X( ) / a : (7) In generalization, the notion of wavelet analysis addresses both, unknown periodicities and non-stationary Therefore, fBm cannot be characterized by the correlation structures. The wavelet analysis is based on time-limited dimension. The main di erence to self-similar processes is elements, the wavelets. By this means, one has the 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 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 wa;b(t) = jaj−1=2w(a−1(t − b)) ; (8) 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, 0.0 0.2 0.4 0.6 0.8 1.0 We use a bi-orthogonal wavelet basis consisting of the collection wj;k(t), where j 2 N is the scaling factor and k 2 Z (set of relative integers) is the translation, together with q(t − k), k 2 Z, where q(t) is a smooth function with a rapid decay. On this basis, x(t) can uniquely be written as x(t) = X ckq(t − k) + X k2Z X dj;kwj;k(t) : j2N k2Z (9) -1.1 -1.0 -0.9 -0.8 -0.7 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), (1 − jtj 0 for jtj 1; otherwise q(t) = and 8> 14 jtj − 12 > > w(t) = < 12 − 34 jtj > :>> 0 for 1 < jtj 2; for jtj 1; otherwise: A main advantage of wavelets is the possibility to calculate the coe cients cjk and djk recursively. The rst trivial step is to take the function q(t) with the same resolution as the sampled signal. We only have to compute the c0k (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 c0k are simply the sampled values of x(t). Then we scale the resolution. We choose the normalizing coe cients such that we replace q(t) by q(t=2)=2. There is only one level of ne fluctuations to add to the smoothed part to recover the signal, and it is de ned by the set of d1;k = d1k. We can now repeat the same procedure to the smoothed part of level 1. This gives two new sets of coe cients c2k and d2k. With our choice of q(t) and w(t), the formulae to obtain the cjk and the djk are simple recursions (Mallat 1989): ckj−−12j−1 + ckj−+12j−1 ; j 1; cjk = 12 ckj−1 + 41 and djk = ckj−1 − cjk: (10) (11) (12) These equations can easily be interpreted in terms of ltering. The cjk are obtained by applying a low{pass lter to the ckj−1 (Eq. 12). On the other hand, the djk being obtained by a di erence between two levels of cjk are in fact the result of a band{pass lter applied to the signal (Eq. 13). This algorithmic scheme is called time-scale analysis or multiresolution analysis (MRA). The decomposition described above is best portrayed in a 2j t { k t - plot of the coe cients djk, called scalogram (Figs. 6-9), which shows the scaling behavior of the radio flux x(t) in dependence on the time location k t. Di erent intensities of the coe cients (amplitudes) are displayed by di erent 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) 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). 4. Scalograms and scalegrams of radio fluxes 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 (Kru¨ger et al. 1994). Using the functions given by Eqs. (10) and (11), we observe that the positive wavelet coe cients djk reflect the burst-like behavior of the radio flux quite well. In x(t) = sin2 Fig. 14. Scalegrams of a fBm realization with = 2:25 preFig. 12. Scalegrams of the event presented in Fig. 2. Dotted sented in Fig. 4. The three di erent scalegrams are calculated line (squares): 0 − 350 s, solid line (diamonds): 350 − 1300 s, for the same intervals as in Fig. 13 dashed line (triangles): 1500 − 1800 s in the last column of Table 1. They agree quite well with Figs. 6-9 the logarithm of the normalized positive wavelet the spectral indices calculated from the structure function. coe cients log2 djk is plotted (scalogram). Summing up the The wavelet transform calculated from the data indiwavelet coe cients over time (index k), we obtain a pic- cates that a rather broad range of time scales from 1 s ture similar to a power spectrum, the so-called scalegram to a few minutes is involved during most parts of the (Figs. 11-14): events (Figs. 6-8, 11-13). The sequences of bright patterns in Figs. 6-8 give an impression of the dominant scales at s(j) = hjdjkj2ik2time interval : (15) di erent times during the bursts. Indications for hierarchic time structures (where di erent, well separated time scales This allows to calculate the spectral index MRA from become dominant) are only very weak and short-lived. For the slope of that part of the scalegram which follows a example, there are short periods around t = 1150 s and power law (Flandrin 1994). Figure 14 shows that, for a t = 1300 s in Fig. 6 and around t = 500 s in Fig. 7 where process with structural similarity to fBm, the lengths of two maxima of the wavelet coe cients do exist. the available time series permit to derive the scaling up For purposes of comparison we have also calculated to about 16 s. The observations in fact show a scaling the scalograms for several kinds of surrogate data, such (power law behavior of the scalegram) from the limit set as white noise, linear colored noise (autoregressive proby the resolution (1 s) up to 16 s (Figs. 11-13) with cesses), and fBm, using comparable data lengths. As exminor deviations for the shortest data sets (< 500 data pected, for white noise the scalogram exhibits complete points). For this range the spectral indices MRA have disorder, whereas periodic features are transformed into been estimated; they are shown in Figs. 15-18 and listed a simple vertical stripe pattern. These stripes are located smaller in the post-impulsive phase, indicating that the at the maxima of the periodic signal. The vertical length emission becomes more random (less correlated) in that of the stripes corresponds to the half width of the period phase (Figs. 15-17). length (see also Fig. 10). In order to check the reliability of the scalegrams, we If we compare the scalogram plots of these di erent have calculated them in each case for 40 slightly di erent models with the data, we nd that the scalogram of an time intervals of equal length. We observe in all cases that fBm with H = 0.625 (Fig. 9) looks very similar to that of the used method is rather robust. The error bars obtained the bursts. This agreement is strengthened by the spectral from these sets of 40 scalegrams for each analyzed time index MRA calculated from the slope of the scalegram, interval are comparable to the symbol size in Figs. 11-14. which nearly equals SF = 2:25. It is important to note that the scalograms from an 5. Energy release models o -Sun position and from a quiet region on the Sun di er signi cantly from those of the bursts. The corresponding We discuss the main ndings of our preceding data analyspectral indices, given in Table 1, show that long-range sis in terms of models for impulsive energy releases in the correlations are missing in these time series (H < 0:5). solar corona. For an overview of the processes involved Next, we analyze the di erent phases of the burst in the radio and related hard X-ray emissions from those separately, i.e. the spectral index MRA is derived for energy release events see, e.g., Benz (1993), Benz & Aschthe pre-impulsive phase, the main phase, and post- wanden (1992). impulsive phase of the bursts shown in Figs. 1-3 sepa- A general approach to energy release events in the rately. Surprisingly, we nd that the spectral indices MRA solar corona is provided by the avalanche model (Lu & change only slightly between these intervals. Thus the Hamilton 1991; Lu et al. 1993; Vlahos et al. 1995). This three phases appear to be structurally analogous, i.e., the statistical model supposes that there is an elementary relative contribution of the di erent time scales remains building block of the energy release processes below the roughly constant. There is a tendency for to be slightly limit of resolution, possessing a threshold which depends only on local conditions, and that the corona is in a state of so-called self-organized criticality, i.e., everywhere and at all times close to onset of an energy release event. This state results from the interplay of external driving (energy input) and more or less localized energy release, where the energy release may organize itself into avalanches of all sizes, since every occurrence of the elementary building block may trigger further occurrences in neighboring regions due to spatial redistribution of stresses and free energy. Interpreting flare events as such avalanches of the elementary process, the model succeeded to reproduce the observed power-law distribution of the occurrence rate of flares versus energy content (Crosby et al. 1993). In order to obtain the power-law, one has to assume that energy release processes in the corona have no characteristic length scale greater than that of the elementary building block (Lu & Hamilton 1991). Then the occurrence rate of avalanches is a monotone decreasing function of energy content. Although the very nature of the energy release process is not addressed by this approach, it basically supposes that coronal energy release processes of all sizes have the same physical nature. This supposition is supported by our nding of structural similarity of preburst, burst, and post-burst phases, respectively. We suppose that the analyzed radio bursts are also composed of elementary building blocks (flux enhancements) not resolved by the available instrumentation, as suggested by their time pro les. One important di erence between the scalegrams (Figs. 11-14) and the flare occurrence rate distribution needs to be clari ed, however. While the latter is derived from a time series (several years long) in which every event occurs separately (i.e., localized in time), each of the former is derived from a time series of one single flare event, which is understood as a superposition of elementary energy release contributions, overlapping in time. Small-scale radio flux enhancements are then underrepresented in the scalograms and scalegrams, since in general they do not stand out clearly in the time pro le of overlapping small-scale and large-scale flux enhancements. The wavelet analysis and the structure function analysis both go beyond the description in terms of event occurrence rates by investigating the structure of single events. This refers to the question whether the energy release in one single flare or burst is composed of a number of avalanches, i.e., is fragmentary. If this is the case, it would be of interest whether the fragments possess a certain size, which would show up as a dominant time scale in the scalegrams. No signs of preferred avalanche sizes (maxima in the scalegrams) were found within the considered range of time scales, which again supports one of the basic assumptions of the avalanche model. We note, however, that there are indications of enhanced occurrence of 0:6 s spikes in hard X-ray time pro les of solar flares (Aschwanden et al. 1995). Work is currently underway to extend the time series analysis of radio bursts into this range of scales. An understanding of the relative contribution of the various time scales to the total radio flux pro le and possible evolutions of the structure of energy release events from the pre-burst to the main and post-burst phase can only be understood in terms of more detailed plasma physical models of the energy release process(es). We restrict ourselves to a brief reference to two currently discussed models, which have both been found consistent with observations of subsecond flux variations in solar bursts (Kru¨ger et al. 1994), viz. the current sheet model and the electric circuit model. We remark that the current sheet model has recently found substantial observational support from X-ray observations of the Yohkoh satellite (Masuda et al. 1994; Tsuneta 1996). The current sheet model involves the formation and dynamical evolution of current laments at small spatial scales (Tajima et al. 1987; Kliem 1995). MHD simulations of current sheet instabilities at high magnetic Reynolds numbers show that the dynamical evolution leads to the pileup of gradients, i.e., the formation of smaller scales than initially present (Schumacher & Kliem 1996). Since the electric eld, which accelerates the particles, peaks in those regions, an enhancement of the small-scale contributions in the flux of fast particles (which is nally reflected by the radio flux) can be expected. Of course, the release of a large amount of energy during the burst phase (a large avalanche) requires the contribution from a large volume (an extended current sheet), hence large temporal scales will signi cantly contribute to the overall time pro le if the release process involves a roughly uniform characteristic velocity (which is the Alfven velocity). The electric circuit model involves a combination of MHD and kinetic instabilities (Zaitsev & Stepanov 1991, 1992) and, consequently, a broad range of time scales. Characteristically, 10 − 100 s scales are predicted by this model for subflares, which is roughly comparable to the maxima in the scalograms obtained here. A more thorough discussion of the two energy release models will become possible if the multiresolution analysis can be applied to burst data with a time resolution t 1 s. Finally we note that the reduced spectral index in the post-burst phases in Figs. 11 and 12 is consistent with an important role of magnetohydrodynamic turbulence in the energy release process. After the main energy release, the energy in the turbulent motions cascades down to smaller scales where it is dissipated into heat, which shows up as a reduced slope in the scalegram. 6. Conclusions We have studied the capabilities of two methods of nonlin ear data analysis (multiresolution analysis and structure function analysis) to characterize complex, non-stationary time series, which are typically observed in solar microwave bursts. Our main ndings are: (1) The wavelet transform permits a local decomposition of Benz A.O., Su. H., Magun A., Stehling W., 1992, A&AS 93, the scaling behavior of the temporal dynamics of the 539 radio flux { in contrast to the usual global methods Crosby N.B., Aschwanden M.J., Dennis B.R., 1993, Solar Phys. of data analysis based on the Fourier transform. This 143, 275 enables the study of non-stationary systems. Di erent Daubechies I., 1992, Ten Lectures on Wavelets, Society for time scales inherent in a series can be resolved, this can Industrial and Applied Mathematics, Philadelphia be done for di erent phases of the series separately. We FFeladnedrrJin., 1P98.,8,1F9r9a4c,talTs.imPele-Sncuamle PrAenssa,lyNseisw aYnodrk Self-Similar have found that the occurrence of an impulsive compo- Stochastic Processes, Byrnes J.S., Byrnes J.L., Hargreaves nent does not imply a signi cant structural change in K.A. & Berry K. (eds.): Wavelets and Their Applications, the properties of scaling in time of the solar microwave NATO ASI Series C 442. Kluwer Academic Publishers, flux. The scalegrams, obtained by averaging over the Dordrecht-Boston-London, pp. 121-142 whole lifetime or a substantial fraction of a burst, do Gu¨del M., Benz A.O., 1990, A&A 231, 202 not show the dominance of a particular time scale in Isliker H., Kurths J., 1993, Int. J. Bifurcation Chaos 3, 1573 the considered range between 1 s and 2 min; dis- Kiplinger A.L., Dennis B.R., Emslie A.G., Frost K.J., Orwig tinct hierarchic time structures and sequences of par- L.E., 1983, ApJ 265, L99 ticular time scales are only short-living phenomena. Kliem B., 1995, in: Coronal Magnetic Energy Releases, Benz This nding is consistent with the idea that the basic A.O. and Kru¨ger A. (eds.), Lect. Notes Phys. 444. Springer, mechanism to generate the solar microwave flux is op- Kru¨BgeerrlinA,.p,.A93urass H., Kliem B., Urpo S., 1987, in: The erating at a broad range of time scales as in the case Sun, Hejna L. and Sobotka M. (eds.), Publ. Astron. Inst. of a turbulence cascade. On the other hand, the time Czechoslovak. Acad. Sci. 66, 245 resolution of the presently available data is limited to Kru¨ger A., Kliem B., Hildebrandt J., Zaitsev V.V., 1994, ApJS 0.5 s, which prevents us from studying the range of the 90, 683 possibly elementary time scales. It is important to note Kurths J., Schwarz U., Witt A., 1995, in: Coronal Magnetic that for all observations investigated here conventional Energy Releases, Benz A.O. and Kru¨ger A. (eds.), Lect. methods, such as correlation and spectral analysis, fail Notes Phys. 444. Springer, Berlin, p. 159 to yield this result. Lu E.T., Hamilton R.H., 1991, ApJ 380, L89 (2) There is no unique well-developed statistical theory for Lu E.T., Hamilton R.H., McTiernan J.M., Bromund K.R., the applied analysis methods yet. To check the relia- 1993, ApJ 412, 841 bility of the estimates obtained, we have applied these Ma6ll7a4t S.G., 1989, IEEE Trans. Pattern Anal. Mach. Intell. 11, techniques (power-law section of both structure func- Masuda S., Kosugi T., Hara H., Tsuneta S., Ogawara Y., 1994, tion and scalegram) also to surrogates chosen to agree Nat 371, 495 in some statistical properties with the data. For solar Meyer Y., Ryan R.D., 1993, Wavelets. Algorithms impulsive mm-wave bursts, the resemblance with the and Applications, Society for Industrial and Applied fractional Brownian motion, governed by a stochas- Mathematics, Philadelphia tic nonlinear evolution equation and characterized by Osborne A.R., Provenzale A., 1989, Physica D 35, 357 scaling exponents H > 0:5, which reflect long-range Priestley M.B., 1981, Spectral Analysis and Time Series. correlations, is evident. Academic Press, London and New York (3) While not distinguishable by the method of power Scargle J.D., 1993, Wavelet Methods in Astronomical Time spectrum, the structure function and the multiresolu- Series Analysis, in Lessi O. (ed.) International Conference tion analysis show clearly di erent scaling exponents oMnetAeporpolilcoagtyi,onPsadoufaTime Series Analysis in Astronomy and for bursts, the quiet Sun, and the sky background. Scargle J.D., Steinman-Cameron T., Young K., et al., 1993, Acknowledgements. We acknowledge fruitful discussions with ApJ 411, L91 L. Vlahos, A. Benz, and P. Maa . We thank Ph. Bendjoya and Schumacher J., Kliem B., 1996, Phys. Plasmas 3, 4703 J.-M. Petit for providing the MRA code. The work of B.K. was Slottje C., 1978, Nat 275, 520 supported by DARA grant 50QL9208. Finally, we thank the Sturrock P.A., Kaufmann P., Moore R.L., Smith D.F., 1984, referee for his helpful criticism. Solar Phys. 94, 431 Tajima T., Sakai J., Nakajima H., et al., 1987, ApJ 321, 1031 Tsuneta S., 1996, ApJ 456, 840 References Urpo S., Pohjolainen S., Ter¨asranta H., 1992, Helsinki University of Technology, Mets¨ahovi Radio Research Station Series A, Report 12 Vigouroux A., Delache Ph., 1994, A&A 278, 607 Vlahos L., Georgoulis M., Kluiving R., Paschos P., 1995, A&A 299, 897 Zaitsev V.V., Stepanov A.V., 1991, AZh 68, 384 Zaitsev V.V., Stepanov A.V., 1992, Solar Phys. 139, 343

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U. Schwarz, J. Kurths, B. Kliem, A. Krüger, S. Urpo. Multiresolution analysis of solar mm–wave bursts, Astronomy and Astrophysics Supplement Series, 309-318, DOI: 10.1051/aas:1998353