Proper Orthogonal and Dynamic Mode Decomposition of Sunspot Data
A. B. Albidah, W. Brevis, V. Fedun, I. Ballai, D. B. Jess, M. Stangalini, J. Higham, G. Verth
rrsta.royalsocietypublishing.org
Research
Article submitted to journal
Subject Areas:
Solar Physics, Mode Decompositionmethods, MHD waves
Keywords:
MHD, POD, DMD, sunspots, waves
Author for correspondence:
Gary Verthe-mail: g.verth@sheffield.ac.uk
Proper Orthogonal andDynamic ModeDecomposition of SunspotData
A. B. Albidah , , W. Brevis , V. Fedun , I.Ballai , D. B. Jess , , M. Stangalini , J.Higham and G. Verth Plasma Dynamics Group, School of Mathematics andStatistics, The University of Sheffield, Hicks Building,Hounsfield Road, Sheffield, S3 7RH, United Kingdom Department of Mathematics, College of ScienceAl-Zulfi, Majmaah University, Saudi Arabia School of Engineering, Pontificia Universidad Católicade Chile, Chile Plasma Dynamics Group, Department of AutomaticControl and Systems Engineering, The University ofSheffield, Sheffield, S1 3JD, United Kingdom Astrophysics Research Centre, School ofMathematics and Physics, Queen’s University, Belfast,BT7 1NN, UK Department of Physics and Astronomy, CaliforniaState University Northridge, Northridge, CA 91330,USA ASI Italian Space Agency, Via del Politecnico snc,00133, Rome, Italy School of Environmental Sciences, Department ofGeography and Planning, University of Liverpool,Roxby Building, Liverpool, L69 7ZT, UK
High resolution solar observations show the complexstructure of the magnetohydrodynamic (MHD) wavemotion. We apply the techniques of POD and DMD toidentify the dominant MHD wave modes in a sunspotusing the intensity time series. The POD techniquewas used to find modes that are spatially orthogonal,whereas the DMD technique identifies temporalorthogonality. Here we show that the combined PODand DMD approaches can successfully identify bothsausage and kink modes in a sunspot umbra with anapproximately circular cross-sectional shape. © The Author(s) Published by the Royal Society. All rights reserved. a r X i v : . [ a s t r o - ph . S R ] O c t r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A ..................................................................
1. Introduction
Analysis of oscillations in sunspot data began in the late 1960’s, see e.g., Beckers and Tallant [1].These authors determined the observational parameters of umbral flashes, a phenomenon thatshows oscillatory behaviour in a sunspot. In the early 1970’s several studies looked at Dopplervelocity oscillations in sunspots. Bhatnagar [2] determined Doppler velocity oscillations with aperiod of the order of − s . Later Beckers and Schultz [3] observed a peak period of around s . Measuring intensity oscillations directly from time lapse filtergram movies, Bhatnagar andTanaka [4] detected periodicities of the order of ± s . Later on, Moore [5] detected Dopplervelocity oscillations of − s and − s in umbral and penumbral regions, respectivelyTo the present day the study of oscillations in sunspots has mainly been carried out byFourier transforming data to provide, e.g. power spectra, either on a pixel by pixel basis orintegrating across a particular Region of Interest (ROI). Although such analysis can providevaluable information, for the identification of coherent structures, e.g. magnetohydrodynamic(MHD) wave modes, in the temporal and spatial domain across a particular ROI, the basic Fouriertransform approach has its limitations. Despite this, one can fine tune a Fourier filter in the spatialand temporal domains to try and identify particular MHD wave modes, as was presented by Jesset al. [6] (hereafter J17) in order to detect a slow kink body mode in a sunspot umbra. In the presentwork we aim to apply the more advanced techniques of Proper Orthogonal Decomposition (POD)and Dynamic Mode Decomposition (DMD) to identify low order MHD wave modes as coherentoscillations across the sunspot umbra, both in the spatial and temporal domains, using the samesunspot data as [6–8]. In the more general solar context, POD has previously been applied todecompose the Doppler velocity of the entire solar disc [9–11] and numerical convection data [12].
2. Observations
The dataset we will analyse has been previously used for studies of running penumbral waves [7],connections between photospheric and coronal magnetic fields [8] and in the detection of anumbral kink mode [6]. The portion of the complete multi-wavelength dataset used in the presentstudy consists of a 75-minute observing sequence of H α images acquired by the Hydrogen-Alpha Rapid Dynamics camera (HARDcam; [13]). The H α time series, which observed theapproximately circular sunspot present within the active region NOAA 11366, were obtainedduring excellent seeing conditions between 16:10 – 17:25 UT on 10 December 2011, with the DunnSolar Telescope (DST) at Sacramento Peak, New Mexico. The sunspot under investigation waslocated at N17.9W22.5 in the conventional heliographic co-ordinate system, or (356 (cid:48)(cid:48) , (cid:48)(cid:48) ) inheliocentric co-ordinates. The filter employed had a full-width at half-maximum of 0.25 Å, whichwas centered on the H α line core at 6562.8 Å. A platescale of . (cid:48)(cid:48) per pixel was used to providea field-of-view size equal to (cid:48)(cid:48) × (cid:48)(cid:48) . On-site high-order adaptive optics [14], post-facto (speckle)image reconstruction techniques [15] and image destretching relative to simultaneous broadbandcontinuum images [16] were implemented to improve the final data products, providing a cadenceof 1.78 s. A sample H α image of the sunspot is displayed in the left panel of Figure 1.
3. Modal decomposition techniques
Following the approach by Higham et al. [17], we are going to employ two techniques to identifylow order MHD modes from the intensity time series. The POD method can be used to identifyMHD wave modes by imposing the criteria that modes are spatially orthogonal. The secondmethod, DMD, assumes a temporal orthogonality of modes. Hence, if observed MHD wave modesdo not have identical frequencies, and this difference is resolved in the frequency domain, thenDMD offers an optimal methodology to identify such modes. r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A ..................................................................
50 100 150 200 250 x y
50 100 150 200 250 x y -3 -2 -1 Hz -10 -8 -6 -4 -2 E Figure 1.
The left panel shows a snapshot from the H α time series with the spatial scale in pixels (one pixel has a widthof . (cid:48)(cid:48) which is approximately 100 km on the surface of the Sun). The middle panel shows the mean intensity of thetime series, the colourbar displays the magnitude of the mean time series, the solid black line shows umbra/penumbraboundary (intensity threshold level 0.85) and the green box ( × pixels) shows the region where we apply ourPOD and DMD analysis. The right panel displays the PSD of the time coefficients of the first 20 POD modes (in log scale).The PSD shows peaks between frequencies . mHz and . mHz (corresponding to periods of 153 - 232s). (a) Proper Orthogonal Decomposition (POD) The POD technique was developed by Pearson [18] as an analogue of the principal axis theorem inmechanics. POD was introduced as a mathematical technique in fluid dynamics by Lumley [19]to identify coherent structures in turbulent flow-fields. In the literature POD takes a variety ofnames depending on the field of application, such as principal component analysis (PCA) andHotelling analysis. Since POD will produce as many modes as there are time snaphots in a dataset,the challenging part of this type of analysis is to identify which of the POD modes actually have aphysical meaning. Hence, for identification of MHD wave modes in the umbral regions of sunspotscare should be taken to compare POD modes with what we should expect from theoretical models,e.g. the MHD wave modes of cylindrical magnetic flux tubes.Let us consider the sequence of ROI intensity snapshots of the sunspot of spatial size X × Y and a time domain of size T . Each of these snapshots can be reorganized in a column matrix W ∈ R N × T , where N = XY and N (cid:29) T , where each column of W will be defined as w i with i = 1 ...T such that W = { w , w , ...w T } . (3.1)There are several approaches to applying POD to a dataset. The classical POD method [20] isperformed by computing the eigenvalues and the eigenvectors of the covariance matrix of thedataset.Another approach is to obtain the POD of W using the optimum low rank approximation [20]and this is known as the Singular Value Decomposition (SVD). Applying the SVD, we obtain W = ΦSC ∗ . (3.2)This decomposition gives the spatial structure of each mode in the columns of the matrix Φ ∈ R N × T ,i.e. φ i with i = 1 ...T and these modes are orthogonal to each other. The temporal evolution ofthe POD modes are given by the columns of the matrix C ∈ R T × T . The particular spatial andtemporal output of the POD presented here is the product of the N two dimensional spatiallyorthogonal eigenfunctions with their associated one dimensional time coefficients. Since PODplaces no restriction on the time coefficients, these can be periodic or aperiodic and the amplitudecan also vary with time. The matrix S ∈ R T × T is a diagonal matrix, and the modes are generallyranked according to their contribution to the total variance of the snapshot series. This contributionis given by the diagonal elements of matrix, λ , by means of the vector λ = diag( S ) / ( N − . r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A ..................................................................
20 40 60 80 100 x y
20 40 60 80 100 x y
20 40 60 80 100 x y -1-0.500.51
20 40 60 80 100 x y
20 40 60 80 100 x y
20 40 60 80 100 x y -1-0.500.51 Figure 2.
The first column displays the spatial structure of the first POD mode with peak power at f = 4 . mHz. Thesecond column displays the spatial structure of the DMD mode that corresponds to the same frequency of f = 4 . mHz.The third column shows the density perturbation of a slow sausage body mode in a cylindrical magnetic flux tube and thedashed circle shows the boundary. In the first and the second columns the solid black line shows the umbra/penumbraboundary as shown in the middle panel of Figure 1 and the dashed circle is used to compare the observations with theflux tube in the third column. The images shown in the two rows are chosen to be in anti-phase, hence, they representdifferent time snapshots. . (b) Dynamic Mode Decomposition (DMD) The DMD technique, first introduced by Schmid [21], is a data-driven algorithm that can extractthe dynamic information of the flow generated by numerical simulations or in a measured physicalexperiment [22]. DMD modes represent the spatial structure of the mode where the associatedeigenvalues give information about the oscillation frequencies of the modes. DMD is a widelyused technique in the field of fluid mechanics, e.g., jet flows [23,24], bluff body flows [25] andvisco-elastic fluid flows [26]). It can therefore also extract information about the coherent spatialstructure of observed MHD wave modes if the modes have distinct frequencies as we show in thisstudy.To apply the DMD technique the time snapshots have to be organized in columns analogouslyto POD, but in two matrices defined as W A = { w , w , ...w τ } and W B = { w , w , ...w T } , (3.3)where W B is shifted by a snapshot of W A such that τ = ( T − . The matrices W A and W B arerelated by a linear operator A ∈ C N × N as W B = AW A . (3.4)DMD is based on approximating the eigenvalues and eigenvectors of the linear operator A withoutactually computing them exactly since for most practical applications the size of A is too large.Using SVD the matrix W A is decomposed as W A = ˜ Φ ˜ S ˜ C ∗ (3.5) r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A ..................................................................
20 40 60 80 100 x y
20 40 60 80 100 x y
20 40 60 80 100 x y -1-0.500.51
20 40 60 80 100 x y
20 40 60 80 100 x y
20 40 60 80 100 x y -1-0.500.51 Figure 3.
The first column displays the spatial structure of the th POD mode with peak power at f = 6 mHz. Thesecond column displays the spatial structure of the DMD mode that corresponds to the same frequency of f = 6 mHz. Thethird column shows the density perturbation of a slow kink body mode in a cylindrical magnetic flux tube and the dashedcircle shows the boundary of the tube. In the first and the second columns the solid black line shows the umbra/penumbraboundary as shown in the middle panel of Figure 1 and the dashed circle is to compare the observations with the flux tubein the third column. The images shown in the two rows are chosen to be in anti-phase, hence, they represent differenttime snapshots. . mHz -10 -8 -6 -4 -2 E mHz -8 -6 -4 -2 E Figure 4.
The left panel displays the PSD of POD 1 mode and it has a peak at . mHz, while the right panel displaysthe PSD of POD 13 mode showing its a peak at mHz. and substituted in Eq. (3.4) to give ˜ Φ ∗ W B ˜ C ˜ S − = ˜ Φ ∗ A ˜ Φ. (3.6)From this we define ˜ A = ˜ Φ ∗ A ˜ Φ, (3.7)where ˜ A ∈ C τ × τ is the optimal low-dimensional representation of A , (note that τ (cid:28) N ) so that wecan calculate the complex eigenvalues, µ i , and associated eigenvectors, z i , of ˜ A , where i = 1 ...τ . r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. To obtain the spatial structure of the DMD modes, we follow the method developed by Jovanovicet al. [24] by calculating a Vandermonde matrix, Q i,j = µ j − i , (3.8)where i = 1 ...τ and j = 1 ...τ . After this operation is completed, the spatial structure of the DMDmodes are obtained from Ψ = W A Q ∗ , (3.9)and the distinct frequencies associated with each these modes are f i = f s arg( z i ) / π, (3.10)where f s is the sampling frequency.
4. Method, results and MHD wave mode identification
Our goal is to use POD and DMD in combination to identify coherent oscillations across thesunspot’s umbra and compare these modes with the MHD wave modes of a cylindrical magneticflux tube predicted from theory.The particular ROI of the sunspot umbra to be studied is shown by the green box in Figure1. Firstly, this ROI is analysed using the POD technique, which ranks modes based on theircontribution to the overall variance. This step is followed by the calculation of the power spectraldensity (PSD) of the POD time coefficient associated with each of these modes. The PSD of the first20 modes, which contains the majority of the energy (96.86 %), show frequency peaks between . mHz and . mHz as shown in the right panel of Figure 1. The PSD of the individual POD modesare then used to determine the dominant frequency, or frequencies if there are a mix of frequencies,associated with a particular POD mode, so that this information could be applied to determine thecoherent spatial structure of modes with distinct frequencies using DMD. If there is no exact matchbetween frequencies, the DMD mode closest to the target frequency is selected.For illustrative purposes we will concentrate on the first branches of the sausage and kink slowbody modes, i.e. modes with only one radial node occurring at the umbra/penumbra boundary.The first POD mode shows the clear azimuthal symmetry of a sausage mode as shown in thefirst column in Figure 2, with a PSD peak at . mHz as shown in the left panel of Figure 4. TheDMD mode that corresponds to the same frequency of . mHz is shown in the second columnin Figure 2. The third column shows the density perturbation of the slow body sausage modefrom the cylindrical magnetic flux tube model. This is important for comparison since the MHDwave modes in a cylindrical flux tube are, by definition, spatially orthogonal. Since the observedumbra is approximately circular, POD, which defines modes by spatial orthogonality, shouldperform well in this particular case study. What is more remarkable is that the DMD technique,which does not have any such criteria, still manages to identify the sausage mode. From boththe POD and DMD analysis there is strong oscillatory power in the penumbra at . mHz andthe penumbral filaments can cleary be identified. Obviously, the idealised cylindrical magneticflux tube model cannot recreate this oscillatory behaviour since it assumes a simple quiescentenvironment without complex fibril structuring. In addition, it is important to state that evenwithin the umbra, disagreement between observations and the eigenmodes of a magnetic cylindercould simply be due to the fact that the observed oscillations are being continually forced and arenot free.The next POD mode that can be interpreted as a physical MHD wave mode is the th modewhich has the azimuthal asymmetry of a kink mode as shown in the first column of Figure 3, witha peak at mHz as shown in the right panel of Figure 4. The DMD mode with frequency of mHz is shown in the second column in Figure 3. Again, for comparison the slow kink body modefrom cylindrical flux tube theory is shown in the third column. Here we can compare these resultswith the previous work of J17. These authors identified a kink mode rotating in the azimuthaldirection by implementing a k − ω Fourier filter ( . − . arcsec − and − . mHz). Hence, r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. the kink mode frequency from POD and DMD is certainly in the same frequency range as thefilter applied by J17. Our analysis reveals that the time coefficients of the POD modes are almostsinusoidal. This is remarkable since POD puts no such condition on these coefficients. Hence,Fourier analysis, which has a sinusoidal basis in the temporal domain, in retrospect was a validapproach. The problem with Fourier analysis is the assumption of a sinusoidal basis in the spatialdomain, since in the cylinder model, the basis functions in the radial direction are Bessel functions.The strength of POD is that it calculates a spatially orthogonal basis, regardless of the geometry ofthe observed waveguide. Also, the further advantage of both POD and DMD over Fourier analysisis that these methods cross-correlate individual pixels in the ROI, in the spatial and temporaldomain, respectively. This is a distinct advantage in identifying a coherent oscillations across thewhole umbra. In agreement with the sausage mode identification in Figure 2, the spatial structureof the POD and DMD modes in the first and second columns of Figure 3 is very similar eventhough the DMD places no restriction on the mode being orthogonal. This further strengthens theargument that the kink mode interpretation is indeed physical.Here we would like to investigate the apparent rotational motion of the kink mode detected byJ17 who constructed a time-azimuth diagram around the circumference of the umbra and estimatedan angular velocity of approximately s − and a periodicity of about 170 s. Physically, therotational motion could be explained by having either (i) a kink mode that is standing in the radialdirection but propagating in the azimuthal direction or (ii) it could be the superpostion of twoapproximately perpendicular kink modes (both standing in the radial and azimuthal directions).Before attempting to recover this rotational motion with the POD and DMD techniques it shouldbe emphasised that the filtering process performed by J17 crudely oversimplified the complexity ofthe swirling "washing machine" motion in the original signal. In particular, the 40 s wide temporalfilter could contain at least least 7 DMD modes. Spatially, the filter effectively divided the umbrainto quadrants. To recreate the apparent rotational motion (or approximate circular polarisation)with POD we need to superimpose at least two spatially perpendicular kink modes with similar,but not necessarily identical, periods. From our analysis this requires the superposition of POD10 (shown on the left panel of Figure 5 and POD 13 shown on the first column on Figure 3). ThePSD of POD 10 has a peak at . mHz as shown on the left panel of Figure 6, while PSD of POD 13has a peak at mHz as shown on the right panel of Figure 4. Both these frequencies lie within thetemporal filter chosen by J17. We can also recreate this rotational motion by superimposing at leasttwo DMD modes. Although DMD modes are not defined to be orthogonal in space, we still findtwo examples of kink modes with DMD that are approximately perpendicular to each other andare also in the same frequency range of J17. These modes correspond to a frequency of . mHz(see the right panel of Figure 5) and mHz shown on the second column of Figure 3. A similartime-azimuth analysis to J17 was performed on the superposition of these two DMD modes alongthe solid black circle shown on the right panel of Figure 5 where the signal was strongest. Thisresulted in an angular velocity of about s − and periodicity of approximately 170 s (see theright panel of Figure 6), consistent with the result of J17.To compare the cylinder model MHD modes with the POD modes from the observationaldata, we also performed a Pearson correlation analysis, calculated on a pixel-by-pixel basis for thesausage (see Figure 2) and kink (see Figure 3) modes using as shown on Figure 7. The result of thecorrelation is a number between 1 and -1, where 1 means that the pixels have a linear correlationwhile -1 denotes a linear anti-correlation. Certainly, there is a better correlation for the sausagethan the kink, but this is not surprising since it is clearly visible from Figure 3 that signal for thekink mode is weaker than for sausage mode (see Figure 2). However, the kink mode stills shows agood correlation in the regions where amplitude is maximum (see Figure 7).
5. Conclusions
All the methods used to identify coherent oscillations across sunspots and pores have theirparticular strengths and weaknesses. We have demonstrated here that a more considered andmulti-faceted approach can be more robust in pinpointing modes that are actually physical. For r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A ..................................................................
20 40 60 80 100 x y -1 -0.5 0 0.5 1
20 40 60 80 100 x y -1 -0.5 0 0.5 1 Figure 5.
Left panel displays POD 10, which is orthogonal in space to POD 13 which is shown on the first column ofFigure 3. The right panel shows the DMD mode with a frequency of . mHz and it is approximately orthogonal in spaceto the DMD mode with a frequency of mHz displayed in the second column of Figure 3. The solid black circle shows thepath of the time-azimuth diagram in Figure 6. mHz -6 -4 -2 E
50 100 150 200 250 300 350
Azimuthal angle (degrees) T i m e ( s ) -1 -0.5 0 0.5 1 Figure 6.
The left panel displays the PSD of POD 10 and it has a peak at . mHz. The right panel shows the time-azimuth diagram after the superposition of two approximately spatially perpendicular kink modes identified with DMD.The white dashed line on the right panel shows gives an apparent angular velocity of about s − consistent with theresult of J17 example, the previous analysis by J17 required fine tuning the Fourier filters in the temporal andspatial domain to reveal the umbral kink mode confirmed by our POD and DMD analysis. Incontrast, POD requires no such filtering, and indeed, such filtering would completely skew theresults. The inherent problem with POD is identifying which modes are physical since this methodproduces as many modes as there are time snapshots. This is where further analysis is requiredas demonstrated in this study and previously by Higham et al. [17]. By calculating the PSD ofeach POD mode the dominant frequency (or frequencies) of each mode can be identified and thesecan be paired with the unique frequencies associated each DMD mode allowing for comparisonbetween the spatial structure of the modes produced by both methods. If there is agreementbetween the spatial structure of both the POD and DMD modes (up to some specified accuracy),then this provides compelling evidence that the mode is indeed physical. To our knowledge, thisis the first time the combined approach of using POD and DMD has been used on sunspot datato identify more than one MHD wave mode. We, therefore, suggest that in combination, PODand DMD could prove to be indispensable tools for decomposing the many possible MHD wavemodes that could be excited in sunspots and pores, especially with the advent of high resolution r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A ..................................................................
20 60 100 x y -1 -0.5 0 0.5 1
20 60 100 x y -1 -0.5 0 0.5 1 Figure 7.
The left panel displays the Pearson correlation between theoretically constructed and observationally detectedsausage mode shown in Figure 2 and the kink mode shown in Figure 3. The positive/negative numbers in the colourbardenote correlation/anti-correlation. observations provided by present and near future ground- and space-based observatories (e.g.Dunn Solar Telescope (DST), Swedish Solar Telescope (SST), The Daniel K. Inouye Solar Telescope(DKIST), Solar-C Space Mission, etc).Data Accessibility.
Authors’ Contributions.
GV and VF initiated the overall research in MHD mode identification. ABA andWB carried out the POD and DMD analysis. VF, IB, GV and ABA provided the theoretical backgroundand physical interpretation of obtained results. DJ and MS provided data sets and participated in datainterpretation. JH provided his expertise in the methodology of mode decomposition. All the Authorsparticipated in discussing the results and editing the draft.
Competing Interests.
The authors declare that they have no competing interests.
Funding.
VF and GV thank to The Royal Society, International Exchanges Scheme, collaboration withChile (IE170301) and Brazil (IES \ R1 \ Acknowledgements. r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. References
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