Modern Data Analysis Methods for Wave and Oscillation Phenomena

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Modern Data Analysis Methods for Wave and
Oscillation Phenomena

• Bernhard Fleck1 Jack Ireland2
• 1ESA Research and Scientific Support Department
• 2L3Com/GSI, c/o NASA/GSFC

Special thanks to James McAteer, Markus
Aschwanden, David Berghmans, Shaun Bloomfield,
Mike Marsh, Scott McIntosh, Ineke de Moortel,
Leon Ofman, Eoghan OShea, Rob Rutten, Tongjiang
Wang
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Outline

• Data / Observations
• Time-Distance Analysis
• Fourier Methods
• Power, Phase, Coherence spectra
• Randomization Test
• Wavelet Techniques
• Travel-Time Analysis
• Other Approaches
• EMD, CEOF, Lomb-Scargle Periodograms,
• Conclusions

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Observations

• Ground (DOT, SST, Sac Peak, MOTH,)

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Observations

• Ground (DOT, SST, Sac Peak, MOTH,)
• TRACE
• SOHO (SUMER, CDS, EIT)
• Future
• Solar-B, SDO, Solar Orbiter

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Oscillations in hot coronal plasma

• The search has been driven by wave-based coronal
  heating theories.
• Few consistent observations in hot material prior
  to 1998.
• EIT and TRACE imaging - 1998, 1999, 2000 - new
  oscillations found.
• SUMER spectroscopy 2002 - MHD wave mode
  conclusively identified.

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Time-Distance Waves in Polar Plumes
DeForest Gurman 1998, ApJ 501, L217
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Time-Distance Waves in Coronal Loops
Berghmans Clette 1999, Solar Phys. 186, 207
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Time-Distance Waves in Coronal Loops
De Moortel, Ireland Walsh 2000, AA 355, L23
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Time-Distance Waves in Coronal Loops

• 80 kms-1 projected velocity
• Amplitude 3
• 1 dominant period above 95 level (5.8 mHz, 170 s)

Marsh Walsh 2006 ApJ 643, 540
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Fourier Methods

• Bracewell 1978
• Brigham 1974
• Groth 1975
• Probability Distributions related to Power
  Spectra (ApJS 286, 29)
• Krijger, Rutten et al. 2002 (AA 379, 1052)

IDL vt fft(v,-1)
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Fourier Methods Power, Phase, Coherence Spectra

• Power spectrum
• Coherence spectrum
• Phase spectrum

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Phase Information

• Dispersion relation
• e.g. acoustic gravity waves in isothermal
  atmosphere (Souffrin, 1966)

H pressure scale height, ? adiab. exponent
?
? By measuring the phase difference ?? between
two layers one can determine kz as a function of
?, and thus the wave type
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Phase SpectrumNa D1 Fe I 5930
Gravity waves
Evanescent waves
Running acoustic waves
Fleck Deubner 1989, AA 228, 506
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2-D Phase SpectraNa D1 Fe I 5930
Theory (dispersion relation)
Observations
Deubner Fleck 1989, AA 213,423
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Is it real or is it noise?

• Randomization test
• NemecNemec 1985, AJ 90, 2317
• Originally developed to test significance of
  periods detected in variable star light curves
• Null hypothesis no periodicity
• Alternative hypothesis the light curve contains
  a periodic function with a given period P.
• Solar applications OShea et al. 2001, AA 368,
  1095

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Is it real or is it noise?

• Randomization test
• NemecNemec 1985, AJ 90, 2317
• Solar applications OShea et al. 2001, AA 368,
  1095)

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Is it real or is it noise?

• Randomization test
• NemecNemec 1985, AJ 90, 2317
• Solar applications OShea et al. 2001, AA 368,
  1095)

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Is it real or is it noise?

• Randomization test
• NemecNemec 1985, AJ 90, 2317
• Solar applications OShea et al. 2001, AA 368,
  1095)

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Randomization Test
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Randomization Test
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Randomization Test
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Randomization Test
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Randomization Test
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Randomization Test
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Randomization Test
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Randomization Test
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Wavelet Analysis - Definition
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Different Mother Wavelets

• Choice of mother wavelet depends on nature of
  signal type of information to be extracted
  from signal
• Frequency resolution larger values of m
• Temporal resolution smaller values of m Paul
  wavelet (De Moortel et al, 2000)

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A Practical Guide to Wavelet AnalysisTorrence
Compo 1998, BAMS 79, 61-78

• ADS search for solar wavelet in abstract
• Up to 1998 76 papers (first one in 1989)
• Since 1999 375 papers
• ADS search for solar fourier in abstract
• Up to 1998 1410 papers (first one in 1913)
• Turner, H.H. MNRAS 73 Sun-Spots and Faculæ, On
  the expression of sun-spot periodicity as a
  Fourier sequence
• Since 1999 735 papers
• 1989-1998 673 papers

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Wavelets A Warning
If I had to give your review I would be nasty to
the wavelet fashion, which very often addresses
glimpses of possible behaviour instead of
collecting enough data to get convincing. Single
wave trains are far too often taken as highly
significant happenings (and the standard
place-consuming four panel diagrams then shown in
full extent, usually greatly overclaiming the
actual information content). My warning would use
the apparently totally uncorrelated successive
wave trains of the 5-min oscillation. The
WhiteCha papers were wavelet phase avant la
lettre and missed the point.
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Wavelet Application to Transversal Coronal Loop
Oscillation

• Horizontal dot-dashed lines from Nakariakov et
  al. (1999) from fitting time series with a
  sinusoidal envelop with exponential decay
• Fourier analysis also gives P ? 200-250 s
• Wavelet scale of the oscillation varies as a
  function of time
• Oscillation period increases linearly
• Any theory that predicts a linearly varying
  oscillation scale?

Ireland de Moortel 2002, AA 391, 339
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Detection of ultra-long-period oscillations in an
EUV filament
- Lomb-Scargle periodogram - Fourier power
spectrum - global wavelet spectrum

• Slow string mode? (Joarder Roberts 1993)

Foullon, Verwichte Nakariakov 2004, AA 427, L5
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Wavelet Analysis
OShea, Banerjee Doyle 2006, AA, submitted
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Wavelet Phase Analysis
Bloomfield et al. 2004, ApJ 617, 623
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Travel-Time Analysis Basic Principle
Courtesy Scott McIntosh
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Travel Time Analysis Measuring both Group and
Phase Velocity
Finsterele, Jefferies et al. 2004, ApJ 613, L185
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Field of View / TopologyMOTH Data
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Travel Time Maps Na V - K VMOTH Data
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Power Maps Na VelocityMOTH Data
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Travel Time Map Na - K
107 hours of MOTH data
Jefferies et al., in prep.
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Travel Time Analysis Results

• Wave propagation behaviour significantly changed
  in and near magnetic field regions
• 3-5 mHz band (evanescent regime)
• Upward propagation with significant time lag in
  strong field regions, while evanescent waves
  (high phase speed, small travel times) in quiet
  sun
• acoustic portals regions of inclined magnetic
  fields ? smaller fcut-off
• f gt 5 mHz (i.e. gt fcut-off)
• small time lag (and in some cases even downward
  propagation) in magnetic field regions, while
  quiet sun showing expected behaviour of upward
  propagating sound waves
• Mode conversion and/or wave reflection at ?1
  surface?
• Power Maps
• Significantly reduced power in magnetic field
  regions, with enhanced power halos at f gt 6 mHz
• Next steps modelling (cf. Carlsson et al.)

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Other methods (1)

• Lomb-Scargle periodogram
• Lomb 1976 Scargle 1982
• good for unevenly sampled data
• well understood confidence level properties
• large literature of application and theory
• Empirical mode decomposition (EMD)
• Huang et al 1998, 1999
• The method gives a decomposition of the signal
  into essentially band-limited components by using
  information from the signal itself instead of
  prescribing basis functions with fixed frequency,
  such as in Fourier analysis, or imposing a
  particular set of basis functions, as is the case
  with wavelet analysis.
• Handles nonstationary data
• Good for filtering - but many empirical aspects
• See Terradas et al., 2004, ApJ 614, 435

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Other Methods (2)

• Multi-fractals
• used to look at bursty type behaviour in time
  series looking for evidence of statistical
  processes (e.g. financial time series, heart
  rhythms, or solar flares)
• Watari, 1996, Solar Phys. 163, 371
• MacAteer et al, in preparation
• Active area of research in many disciplines, but
  problems in implementation and interpretation.
• Complex Empirical Orthogonal Functions (CEOF)
• Observed oscillations occur in space and time -
  the signal is decomposed into spatial components
  (eigenfunctions) that have a time varying
  amplitude.
• The eigenvalues describe how strong each
  component is.
• See Terradas et al., 2004, ApJ 614, 435
• Noise properties not well understood - unless you
  know better!

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Many, many other methods exist - for example.

• Entropy approach
• The correct period of the time series orders the
  data in the best possible way - this leads to the
  information (Shannon) entropy approach to
  identifying periods
• Cincotta et al., 1995, ApJ 449, 231
• Expansion into orthogonal polynomials
• better for non-sinusoidal signals
• Schwarzenberg-Czerny, 1996, ApJ 460,L107
• Phase Dispersion Minimization
• Suited for nonsinusoidal time variations covered
  by only a few irregularly space observations
• Stellingwerf, 1978, ApJ 224, 953
• Singular Spectrum Analysis
• separates periodic, chaotic, and random
  components
• Watari, 1996, Solar Phys. 186, 413

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Conclusions 1/2

• Forthcoming instruments need significant
  improvements in spatial resolution
• Solar-B ?
• Solar Orbiter ?
• Detection of oscillation is not the end, but just
  the start
• Whats next?
• Need to move to more statistical studies
• what are the properties of this "class of
  oscillations" rather than one particular wiggle
  seen in one particular data set
• Coronal seismology is in its infancy
• shows a lot of promise
• relies on data analysis methods getting as much
  information out of the data as it can safely give
  you
• should be coupled to the unambiguous
  identification with wave modes

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Conclusions 2/2

• Where are the waves that might be responsible for
  coronal heating? 
• If they remain unobservable, why is that? 
• Currently observable wave modes tell us more
  about the state of the corona than they do about
  coronal heating. 
• What instruments are required in order to see the
  waves that may cause coronal heating? 
• Reconnection theory is a big driver for getting
  better spatial resolution - shouldn't wave theory
  be a big driver for getting better spectral
  resolution (identification of wave modes,
  measuring travel times between different layers
  of the atmosphere)?
• Look at whats happening in other fields
• cf. success stories of wavelets or randomization
  test
• If you want give your citation index a boost
  Implementation in IDL SolarSoft (cf. Torrence
  Compo)