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Global Helioseismology 1: Principles and Methods

Rachel Howe, NSO

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Introduction to Global Helioseismology

• What is helioseismology?• A bit of early history• Basics

– p-modes and g-modes– Spherical harmonics and their labeling

• Observations– Instrumentation– Networks and spacecraft– Time series– Spectra

• Methods– Peak finding– Inversions

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What is helioseismology?

• Helioseismology utilizes waves that propagate throughout the Sun to measure its invisible internal structure and dynamics.

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History

• Discovered in 1960 that the solar surface is rising and falling with a 5-minute period

• Many theories of wave physics postulated:– Gravity waves or acoustic

waves or MHD?– Where was the region of

propagation?• A puzzle – every attempt to

measure the characteristic wavelength on the surface gave a different answer

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The puzzle solved

• Acoustic waves trapped within the internal temperature gradient predicted a specific dispersion relation between frequency and wavelength

• A wide range of wavelengths are possible, so every early measurement was correct – result depended on aperture size

• Observationally confirmed in 1975

• 5,000,000 modes, max amplitude 20 cm/s

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Inside the Sun

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Three types of modes

• G(ravity) Modes – restoring force is buoyancy – internal gravity waves.– Amplitude vanishes at the surface

• P(ressure) Modes – restoring force is pressure.– Amplitude peaks at the surface.– Turning point depth/phase speed decreases with l.

• F(undamental) Modes – restoring force is buoyancy modified by density interface – surface gravity waves.– Can usually be thought of as n=0 p modes.

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p-mode anatomy

• A p mode is a standing acoustic wave.

• Each mode can be described by a spherical harmonic.

• Quantum numbers n (radial order), l (degree), and m (azimuthal order) identify the mode.

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Spherical Harmonics

• The harmonic degree, l, indicates the number of node lines on the surface, which is the total number of planes slicing through the Sun.

• The azimuthal number m, describes the number of planes slicing through the Sun longitudinally.

•Picture credits: Noyes, Robert, "The Sun", in _The New Solar System_, J. Kelly Beatty and A. Chaikin ed., Sky Publishing Corporation, 1990, pg. 23.

l=6, m=0 l=6, m=3 l=6, m=6

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Mode in Motion

• Rotation lifts degeneracy between modes of same l, different m.

• Prograde and retrograde modes have different frequencies.

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Turning points

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Duvall law• Modes turn at depth

where sound speed = horizontal phase speed = ν/ℓ

• So, all modes with same ν/ℓ must take same time to make one trip between reflections

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Observational Requirements

• Typical p-mode amplitudes around 1-10cm/s• Need to measure velocity of solar surface to

parts in 105.• Modes have periods around 5 minutes, so

typical cadence 30-60s gives adequate Nyquist frequency.

• Need to observe for > 1 month to get good frequency resolution for medium-l modes.

• Observations should be as nearly continuous as possible.

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Why we need continuous observations

• The sun sets at a single terrestrial site, producing periodic time series gaps

• The solar acoustic spectrum is convolved with the temporal window spectrum, contaminating solar spectrum with many spurious peaks

• In turn, this can distort the science results

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How to get continuous observations

• South pole (in Austral Summer)– Harsh conditions.– Weather.– Only possible for part of year

• Global network– Ideally at least six stations to provide overlap.– Can get 80-90% fill if well funded and maintained.– Data can be mailed home.– Data need to be combined.– Still observing through atmosphere.

• Spacecraft– No atmosphere, so cleaner measurements.– Can get nearly 100% coverage from one instrument.– Expensive, hard/impossible to repair.– Telemetry can be costly (DSN).

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BiSON

• 6-site network of single-pixel instruments, data since 1976, completed 1992.

• Modes up to l=4• Run by University

of Birmingham, UK

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The GONG(+) network

• Six stations around the world for continual coverage.

• 256x256 pixels 1995-2001

• 1024 pixels since 2001

• Run from NSO Tucson.

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MDI aboard SOHO

• ESA/NASA spacecraft orbits the Lagrange point between Sun and Earth, a million miles away.

• Many instruments, of which MDI is one of 3 for helioseismology.

• MDI has 1024×1024 pixels, but usually bins down to 256×256

• Operating since 1996.

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Coming Soon: HMI aboard SDO• HMI (Helioseismic and

Magnetic Imager) aboard SDO (Solar Dynamics Observatory), due to launch 2008.

• Earth Orbit.• 4096x4096 pixels, all the

time.

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Observing p-modes• Doppler measurements at the surface . . .

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Spatial Harmonic Transform

X

X

X

=

=

=

Σ

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Temporal Fourier TransformTime Series Power Spectrum

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2d spectrum (l- diagram)

Degree l

Fre

quen

cy

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m-diagram

Differential rotation lifts degeneracy between different m modes of same l.

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Curved shape shows Differential Rotation

Multiple ridges due to leakage

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Part 2

• Peak Finding– Statistics– Asymmetry– Leakage

• Inversion Principles and Techniques– Eigenfunctions and kernels – The inversion problem– RLS and OLA techniques– Averaging kernels – Errors– Error correlation functions– Structure inversions

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Peakfinding

• Non-linear optimization• Modes are stochastically

excited.• Spectrum can be

considered as ‘limit’ spectrum multiplied by noise distributed as 2 with 2 degrees of freedom. (N.B. not Gaussian.)

• Standard least-square fits not appropriate.

a,b show two different ‘realizations’ for short observations: c shows result for longer observations; d is limit spectrum.

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Peakfinding

• Instead of 2, minimize ‘log-likelihood’ function:

• where M is model, O observations, a is vector of parameters.

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Peak profile

• Standard model is a Lorentzian profile.

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Granulation and Excitation

• The oscillations are excited by solar granulation, which generates a randomly excited field of damped Helmholtz oscillators.

• Excitation comes from downward plumes in intergranular lanes.

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Velocity and Intensity

• Measurements can be made in brightness (intensity) or Doppler velocity

• Intensity from ground can be noisy.

• Different information from each.

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Excitation Puzzles

Line asymmetryV-I frequency offset

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Asymmetry

• In reality, the observed peaks in the spectrum have some asymmetry, which is understood in terms of noise correlated with the oscillations.

• Observations in velocity and intensity show different asymmetry behavior.

• This can lead to peaks apparently having different frequencies in velocity and intensity spectra.

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Asymmetric Peak Profile Model

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Leakage• Because we see only part of

the Sun’s surface, the spherical harmonics are not orthogonal.

• Therefore, we cannot completely isolate the different (m,l) spectra; each spectrum contains power from adjacent ones, which has to be taken into account in fitting.

• The problems are most severe when the peaks overlap the leaks.

• The leakage characteristics need to be calculated for many fitting schemes.

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IntroductionThe leakage matrix is calculated by emulating the processing of an image through the GONG processing pipeline, using the desired (l’,m’) spherical harmonic pattern instead of the solar velocity image.

IntroductionThe leakage matrix is calculated by emulating the processing of an image through the GONG processing pipeline, using the desired (l’,m’) spherical harmonic pattern instead of the solar velocity image.

Remap to x,

Apodize

SHT for l, m

FFT

(R+I)/2Leakage coefficient Llml’m’

2 = power of l’,m’ leak in l, m spectrum

l, m power spectrum with l’, m’ leaks

Leakage coefficients

Time series

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Leakage rules-of-thumb

• Leaks with l+m odd vanish.

• Problematic cases are those where leaks not resolved from wanted peaks (≤)– ‘m-leaks’: l=0, m=±2; .Hz– ‘n-leaks’: l=1, n=±1, ±2 if overlapping.– ‘l-leaks’: l=±1, m=±1 if overlapping. (High l).

• It’s more complicated than that.

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Inversions

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Inversions

• Modes are reflected due to density variations.

• The lower the l, the fewer surface reflections, and the deeper the mode penetrates.

• Combining information from different modes lets us build up a picture of properties at different depths.

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l=50,m=0

Inversions

• Modes of different m cover different latitude ranges, giving latitudinal resolution.

m=45

m=50

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The (rotation) inversion problemKernel

Averaging Kernel

Coefficients to be found

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Regularized Least Squares – fit the model to the data!

Minimize 2

Regularization

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Subtractive Optimally Localized Averages – Optimize the Kernel!

Specify desired averaging kernel shape T, and minimize

Regularization

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Inversion Errors

For input data with independent errors,

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RLS OLA

OLA

RLS

Close-Up

2-D Rotational Averaging Kernels

(1 s.d. uncertainties on inversion are indicated in nHz, for a typical MDI dataset)

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Choosing Tradeoff Parameters

• Compromise between errors and localization.

• Heavier regularization gives smaller errors, poorer resolution.

(For data with uniform errors

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Error Correlation Functions

• The errors in the inversion result are not independent, even if the input data are.

• Just how correlated are errors between two locations in an inversion result?

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Structure Inversions

• Not linear, but can use variational principle for small differences from a model.

• Fundamental variables are p, and the adiabatic exponent .

• Because the Sun’s mass is fixed, these are not all independent, and the problem can be reduced to variable pairs, for example, (c2,) or (u, Y) where u=p/ and Y is the helium abundance.

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Structure Inversions

• Linearized 1d version, after taking difference from model values.

Surface TermError

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Kernel for c2

Kernel for ρ

Kernels for sound speed and density