A&A 592, A153 (2016) Astronomy DOI: 10.1051/0004-6361/201628306 & c ESO 2016 Astrophysics Empirical mode decomposition analysis of random processes in the solar atmosphere D. Y. Kolotkov1, S. A. Anfinogentov1, and V. M. Nakariakov1; 2; 3 1 Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, CV4 7AL, UK e-mail: [email protected] 2 School of Space Research, Kyung Hee University, Yongin, 446-701 Gyeonggi, Korea 3 Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, 196140 St Petersburg, Russia Received 13 February 2016 / Accepted 10 June 2016 ABSTRACT Context. Coloured noisy components with a power law spectral energy distribution are often shown to appear in solar signals of various types. Such a frequency-dependent noise may indicate the operation of various randomly distributed dynamical processes in the solar atmosphere. Aims. We develop a recipe for the correct usage of the empirical mode decomposition (EMD) technique in the presence of coloured noise, allowing for clear distinguishing between quasi-periodic oscillatory phenomena in the solar atmosphere and superimposed random background processes. For illustration, we statistically investigate extreme ultraviolet (EUV) emission intensity variations observed with SDO/AIA in the coronal (171 Å), chromospheric (304 Å), and upper photospheric (1600 Å) layers of the solar atmo- sphere, from a quiet sun and a sunspot umbrae region. Methods. EMD has been used for analysis because of its adaptive nature and essential applicability to the processing non-stationary and amplitude-modulated time series. For the comparison of the results obtained with EMD, we use the Fourier transform technique as an etalon. Results. We empirically revealed statistical properties of synthetic coloured noises in EMD, and suggested a scheme that allows for the detection of noisy components among the intrinsic modes obtained with EMD in real signals. Application of the method to the solar EUV signals showed that they indeed behave randomly and could be represented as a combination of different coloured noises characterised by a specific value of the power law indices in their spectral energy distributions. On the other hand, 3-min oscillations in the analysed sunspot were detected to have energies significantly above the corresponding noise level. Conclusions. The correct accounting for the background frequency-dependent random processes is essential when using EMD for analysis of oscillations in the solar atmosphere. For the quiet sun region the power law index was found to increase with height above the photosphere, indicating that the higher frequency processes are trapped deeper in the quiet sun atmosphere. In contrast, lower levels of the sunspot umbrae were found to be characterised by higher values of the power law index, meaning the domination of lower frequencies deep inside the sunspot atmosphere. Comparison of the EMD results with those obtained with the Fourier transform showed good consistency, justifying the applicability of EMD. Key words. Sun: atmosphere – Sun: photosphere – Sun: chromosphere – Sun: corona – sunspots – methods: data analysis 1. Introduction physical nature, for example, they can be caused by instrumental artefacts or by random processes operating in the solar atmo- The solar atmosphere evidently shows a wide range of peri- sphere. Interestingly, in the majority of previous studies these odicities detected throughout the whole electromagnetic spec- modes were usually ignored and simply disregarded in analysis. trum. They have different physical nature and causes, and their However, recent works have shown that coloured noises can be periods vary from a fraction of second up to several years, recognised in solar and stellar flare light curves with the Fourier and even to centuries. Some examples of the oscillatory pro- power spectrum, and accumulate a significant part of a signal’s cesses responsible for the periodicities are the 11 yr solar spectral energy. They are manifested as a power-law-like Fourier cycle, the so-called quasi-biennial oscillations, the helioseis- power spectra, and seem to be intrinsic features of many obser- mic variations (see Hathaway 2010; Bazilevskaya et al. 2014; vational data sets detected by different instruments. For exam- Christensen-Dalsgaard 2002, and references therein for recent ple, Inglis et al.(2015) considered flare signals exhibiting QPP comprehensive reviews), magnetohydrodynamic waves and os- detected with the PROBA2 (Large Yield Radiometer), Fermi cillations in different plasma structures of the solar atmosphere (Gamma-ray Burst Monitor), Nobeyama Radioheliograph, and (see De Moortel & Nakariakov 2012; Liu & Ofman 2014; Yohkoh (HXT) instruments. They found that the majority of Jess et al. 2015), and various quasi-periodic pulsations (QPP) cases considered could be described by a power law in the appearing in solar flare light curves (Nakariakov & Melnikov Fourier power spectra. Signatures of strong noisy components 2009; Kupriyanova et al. 2010; Simões et al. 2015). having power-law-like Fourier power spectral densities were In addition to clear oscillations, broadband modes usually also detected in Gruber et al.(2011), where the RHESSI and associated with noise often appear in solar signals of various Fermi (Gamma-ray Burst Monitor) observations of solar flares types too. These noisy components may have basically different were considered. Application of the wavelet transform modulus Article published by EDP Sciences A153, page 1 of9 A&A 592, A153 (2016) maxima method showed the multifractal spectra of the tempo- made in Franzke(2009), revealing noise-like properties of the ral variation of the X-ray emission in solar flares (McAteer et al. Earth’s climate data. 2007). In this paper we extend the studies of Wu & Huang(2004) The interest in the coloured noise in the solar atmosphere and Franzke(2009), and reveal similar empirical properties of is connected with its possible link with various dynamical phe- coloured noises in EMD, allowing for arbitrary indices in power nomena. For example, in the case of QPP in flares a frequency- law dependences of their power spectral densities in application dependent noise may be associated with repetitive magnetic re- to the solar atmosphere. We show that coloured noises can be connection (e.g. Bárta et al. 2011). In the case of the quiet Sun, successfully described by the chi-squared distribution too. How- the evolution of the noise with height may reveal the physical ever the parameter of the distribution function, the number of processes responsible for the generation, dissipation and evolu- degrees of freedom (DoF), needs to be adjusted accordingly tion of magnetohydrodynamic waves, turbulence, and episodic for each sort of noise. For illustration of the reliability of the energy releases. Recently, Ireland et al.(2015) studied mean method, we adapt this EMD-based technique for analysing so- Fourier spectra of various regions of the solar corona observed at lar extreme ultraviolet (EUV) data sets obtained with SDO/AIA, testing them in the manner described above for the presence of the 171 Å and 193 Å wavelengths, and found that they can be de- randomly distributed dynamical processes. Historically, obser- scribed by a power law at lower frequencies, tailing to a flat spec- vations of the oscillatory processes in the atmosphere of the Sun trum at higher frequencies, plus a Gaussian-shaped contribution attract a great interest among the research community. In particu- specific for different regions of the corona. Also, understanding lar, the high-frequency tails of solar dynamical spectra, obtained of the noise is important for the development of automated detec- from the lower layers of the solar atmosphere, were found to tion techniques (e.g. Nakariakov & King 2007; Sych et al. 2010; vary significantly with the height and magnetic properties of the Ireland et al. 2010; Martens et al. 2012). region (see, e.g. Evans et al. 1963; Orrall 1966; Woods & Cram In addition to the Fourier and wavelet techniques, another 1981; Deubner & Fleck 1990). In this paper we use the data ob- spectral method being actively used for the analysis of solar tained with the modern SDO/AIA instrument. Due to the ad- signals is the Hilbert-Huang Transform technique developed in vanced combination of this data’s spatial and temporal resolu- Huang et al.(1998), Huang & Wu(2008). It is based upon the tions and high values of the signal-to-noise ratio, it allows us to empirical mode decomposition (EMD) of the signal of interest investigate the higher coronal altitudes of the solar atmosphere into the basis derived directly from the data by iterative search- in the EUV band and directly compare their dynamical spectra ing for the local time scale naturally appearing in the signal. Be- with the chromospheric and photospheric layers. We show that ing not restricted by an a priori assignment of the basis func- application of this EMD-based noise-test to SDO/AIA data sets tion, EMD operates adaptively and, hence, is essentially suitable revealed that they mainly consist of random signals represented for processing non-stationary and non-linear time series typical by a combination of the white noise at shorter-period spectral for the solar signals. These unique properties of EMD attracted components and the reddish noises at longer periods. a growing interest in the application of this technique to anal- ysis of dynamical phenomena on the Sun. This technique has already been successfully applied to solar signals. For exam- 2. Methodology and properties of coloured noises ple, anharmonic and multi-modal structures of solar QPP were in EMD revealed with EMD in Nakariakov et al.(2010), Kolotkov et al. Power spectral density S of coloured noises as a function of (2015b), detailed two-dimensional information about a propa- α gating and a standing wave in a coronal loop was obtained with frequency, f can be written as S = C= f , where C is a con- EMD in Terradas et al.(2004), periodicities associated with the stant which can be reduced to unity by appropriate normalisa- 11 yr solar cycle were investigated with EMD in Kolotkov et al.