Just how bursty is X-ray data?

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Number: 56
1st Author: R.T. James McAteer
2nd Author:
Published: 23 January 2007
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When confronted with lightcurves of RHESSI data for the first time, the natural reaction is one of consternation over their intricate, complicated nature. The lightcurves are far from smooth and regular. In contrast, they tend to be spiky and bursty, with rapid changes over small time periods. As RHESSI has excellent sensitivity down to the smallest timescales (4 sec to avoid the temporal modulation created by the image structure), the undoubtedly rich, detailed physics taking place in solar flares is reflected in these lightcurve data. Several key features of the standard flare model are reflected in this complexity and the scientist is left trying to solve two main problems.

How can I separate the real data from the noise, without losing scientifically useful information? How many of these bursts and spikes are real flare related events and how many are non-solar (e.g., background, instrumentation or atmospheric)? What should I study, and how can I omit the noise? How can I then describe all this remaining complexity in a meaningful manner? What does it mean when a lightcurve in one flare is smoother than a lightcurve from a different flare? Does the spiky nature of the lightcurve change with time throughout a flare, or at different energies? Fortunately other people working in completely unrelated fields of study may have given us the tools to answer these questions.

Why are the the lightcurves bursty?

There are numerous parts of the `standard flare' model which naturally give rise to bursty lightcurves from flares. At the small size scales necessary for magnetic reconnection, numerous energy release events are probably occurring. We are, in all likelihood, only observing a spatial and temporal integration of these events. Under this explanation, each spike is a new event and the resulting emission is superimposed on the decay or rise phase of some neighboring event. This could explain the departure from smoothness. In a perfect world where we could observe one isolated reconnection event, would we find a smooth lightcurve? The answer is probably no as almost all the proposed acceleration mechanisms (intermittent DC electric fields, stochastic acceleration, shock acceleration) are intrinsically complex in their own nature.

So now we have a picture of spatial and temporal integration of events, which, even when isolated, look bursty. In addition to this is a third integration, the Neupert effect. Emission at lower X-ray energies closely resembles the time integration of the higher energy X-ray emission. The inevitable observational result is that higher energy emission tends to be much more bursty than the lower energy counterpart for any flare (so GOES lightcurves appear much smoother than RHESSI lightcurves)

Figure 1. Lightcurves in all energy bands are bursty, The bursty nature is more prevalent at higher energies and varies throughout the flare

The Sandpile model

Numerous authors have used this bursty nature (or 'departure from smoothness') of the lightcurves in modeling. Some authors use the actual bursts, interpreted as pulsed particle injection. Others remove the bursty component and study the remaining smoother curves to look for oscillations as signature of waves. The self-organized criticality model (SOC) has achieved a lot of success in trying to describe the global observational features from an unstable starting point. A good conceptual realization of SOC models can be gained by thinking of someone continually adding a grain to a pile of sand. The pile if initially stable then just adding one grain at a time does not seem to make much difference. But the system is intrinsically unstable and is slowly building up the potential to produce an avalanche. At some stage, one single grain too many will cause neighboring grains to shift. This will set off an unstoppable chain of collapsing grains, an avalanche will occur, and the entire system will have been rearranged. Any time series of this event (e.g. height of pile, or number of moving grains) will exhibit the bursty, spiky appearance of RHESSI lightcurves

These types of models have achieved success in fields as diverse as economics, medicine and geology via the combined use of multiscale and multifractal work in an algorithm now known as wavelet transform modulus maxima . It is the advances in these fields that we have unashamedly adopted for studying the complexity of solar flare X-ray lightcurves.

What is multifractality?

The mathematical details of multifractality are best left to other websites. Conceptually, it can be thought of by starting off with data from a smooth, predictable, world. Lightcurves from this imaginary regime can be well approximated by only a few terms of a Taylor series and it is relatively simple to manipulate (e.g., differentiate) the data. In the real world however, data tend to exhibit fractal features, which are describable as non-differentiable singularities. The strength of a singularity is described by its Hurst exponent , H. Quantifying the singularity in this way makes it possible to describe different types of variation as regimes of different singularity strengths. Applied to RHESSI data, we can separate and discount any singularity with an exponent representative of noise (see question 1 above).

Figure 2. The Hurst and Holder exponents. In most common cases, the Hurst and Holder exponents are related simply by h=H-1. The various regimes of each exponent correspond to different types of behavior.

Different singularities in the lightcurve may have different Hurst exponents, so a frequency histogram can be built up, describing the probability distribution function of each H value - the multifractal spectrum. Numerous algorithms have been suggested to calculate the multifractal spectrum of a signal, and recently the 'wavelet transform modulus maxima' (WTMM) algorithm has achieved a lot of success. The use of a wavelet ensures the time localization to address question 2 above, the use of the modulus maxima makes the algorithm robust and efficient. For the WTMM, instead of the Hurst exponent (which strictly only applies to monofractal series), we use the Holder exponent, h, involving an integration of the signal and thereby creating an increase of 1 when compared to the Holder exponent (h=H-1). Furthermore, instead of calculating each singularity separately, the WTMM approach take a global statistical approach of calculating the Hausdorff dimension, a needed characterization, for each h value.

The multifractality of solar flare X-ray emission

Figure 3. The multifractal spectrum of the 25-50 keV emission. The x-axis is Holder exponent and the y-axis is Hausdorff dimension (filling factor) of each exponent. Hence the most common Holder exponent is at the peak of the graph. The q values refer to the details of the algorithm explain in more detail in the Astrophysical Journal paper.

The multifractal spectrum of the 25-50 keV emission is displayed in Figure 3. The peak of the curve (the Holder exponent with largest Hausdorff dimension), hH = 0.95±0.06 exists in the "persistent walk" regime and can be thought of as the most characteristic exponent of the data. As this occurs at D(hH) = 1.02±0.04, this basically exists throughout the signal. The singularity spectrum extends from h = 0.35±0.02 to h = 1.46±0.20. The left leg of the multifractal spectrum extends into the "anti-persistent walk" range. This suggests contamination of a smooth regular component with a more bursty component in this energy range (we note this is the typical break energy of the thermal / non-thermal components in X-ray spectroscopy)

Figure 4. The multifractal spectrum of each energy band. At lower energies, the emission is persistent, and gradually moves into the anti-persistent range at higher energies.

Figure 4 shows the multifractal spectrum of each energy band. The 'thermal' energy bands (3-6 keV, 6-12 keV, 12-25 keV) are at the top. These spectra peak at a large Holder exponent and exist entirely in the persistent walk regime. This suggests the physical cause of the emission has a long term memory. This is expected if the emission is from thermal energy evaporated into flare loops during the impulsive phase with subsequent cooling.

At higher energies (25-50 keV, 50-100 keV) the right leg of the multifractal spectrum remains in the persistent walk range. However the spectrum peaks at a much smaller Holder exponent, and extends into the anti-persistent walk range for the left leg. Although the 50-100 keV spectrum peaks at the same value as the 25-50 keV plot, the range is shifted to lower values. This is further evidence of the increasing bursty nature at higher energies. The non-symmetric nature of the 50-100 keV spectra is also typical of a system where two or more processes are at work; here a weak thermal component of the emission is described by the rapidly decaying right leg of the spectrum, and a stronger non-thermal component of the emission is described by the the more gentle decay of the left leg.

At higher energies (100-300 keV), the multifractal spectrum peaks in the anti-persistent walk range. At 300-800 keV, the spectrum exists entirely inside the anti-persistent regime. Mathematically, these lightcurves therefore have a short memory and this may be the direct result of pulsed electron beams, which are often modeled as bursts of uncorrelated behavior.


The study of singularities and complexity is well-studied across multiple subject areas. By borrowing their approaches and ideas, and applying them to solar physics, we can gain new insight into the complex, singular behavior we may be observing on the Sun. Currently we have only applied this to emission from one flare, integrated over the solar disc. A larger flare sample, and integrating over spatially localized areas, may give us more clues to the physics behind this bursty emission. We encourage anyone to read our Astrophysical Journal paper and try the code themselves by downloading the software and example data.

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