1 Global Helioseismology 1: Principles and Methods Rachel Howe, NSO

2 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

3 What is helioseismology? Helioseismology utilizes waves that propagate throughout the Sun to measure its invisible internal structure and dynamics.

4 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

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

6 Inside the Sun

7 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.

8 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.

9 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=0l=6, m=3l=6, m=6

10 More spherical harmonics

11 Mode in Motion Rotation lifts degeneracy between modes of same l, different m. Prograde and retrograde modes have different frequencies.

12 Spherical Harmonic Animations

13 Turning points

14 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

15 Observational Requirements Typical p-mode amplitudes around 1-10cm/s Need to measure velocity of solar surface to parts in 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.

16 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

17 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).

18 BiSON 6-site network of single-pixel instruments, data since 1976, completed Modes up to l=4 Run by University of Birmingham, UK

19 The GONG(+) network Six stations around the world for continual coverage. 256x256 pixels pixels since 2001 Run from NSO Tucson.

20 Better resolution …

21 … lets us access higher l modes

22 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.

23 Coming Soon: HMI aboard SDO HMI (Helioseismic and Magnetic Imager) aboard SDO (Solar Dynamics Observatory), due to launch Earth Orbit. 4096x4096 pixels, all the time.

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

25 Spatial Harmonic Transform X X X = = = Σ

26 Temporal Fourier Transform Time Series Power Spectrum

27 2d spectrum (l- diagram) Degree l Frequency

28 m-  diagram Differential rotation lifts degeneracy between different m modes of same l.

29 Curved shape shows Differential Rotation Multiple ridges due to leakage

30 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

31 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.

32 Alternative Approaches If we average enough spectra from the same limit spectrum, the statistics tend back to Gaussian/Normal. –BUT, everything varies (with time, frequency, m), so hard to find enough spectra to average. Can also apply smoothing schemes (running mean, multitaper, Gaussian denoising). –BUT the statistics are more complicated, and peaks can be distorted.

33 Peakfinding Instead of  2, minimize ‘log-likelihood’ function: where M is model, O observations, a is vector of parameters.

34 Peak profile Standard model is a Lorentzian profile.

35 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.

36 Velocity and Intensity Measurements can be made in brightness (intensity) or Doppler velocity Intensity from ground can be noisy. Different information from each.

37 Excitation Puzzles Line asymmetry V-I frequency offset

38 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.

39 Asymmetric Peak Profile Model

40 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.

41 Introduction The 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. Introduction The 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,  ApodizeSHT for l, m FFT ( R+I )/2 Leakage coefficient L lml’m’ 2 = power of l’,m’ leak in l, m spectrum l, m power spectrum with l’, m’ leaks Leakage coefficients Time series

42

43 Leakage Equations Mask M from Y l m,apodization Distance  from disk center

44 The power from mode (l’,m’) leaking into the (l,m) spectrum is given by L 2 lml’m’ where L lml’m’ =(c’ lml’m’+ c lml’m’ )/2.

45 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.

46 Self-leakage The fraction of the power of a given mode that is seen in its own spectrum. Lowest at low m Falls off at higher l for GONG classic

47 Relative power of 1 st m-leak Note noisy GONG classic result!

48 Inversions

49 Modes of different l sample different depths Modes are reflected due to density variations. The lower the l, the fewer surface reflections, and the deeper the mode penetrates.

50 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.

51 l=50,m=0 Inversions Modes of different m cover different latitude ranges, giving latitudinal resolution. m=45 m=50

52 The (rotation) inversion problem Kernel Averaging Kernel Coefficients to be found

53 Regularized Least Squares – fit the model to the data! Minimize 22 Regularization

54 Subtractive Optimally Localized Averages – Optimize the Kernel! Specify desired averaging kernel shape T, and minimize Regularization

55 Inversion Errors For input data with independent errors,

56 RLS OLA RLS Close-Up 2-D Rotational Averaging Kernels (1 s.d. uncertainties on inversion are indicated in nHz, for a typical MDI dataset)

57 Examples of averaging kernels

58 Choosing Tradeoff Parameters Compromise between errors and localization. Heavier regularization gives smaller errors, poorer resolution. (For data with uniform errors 

59 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?

60 Duvall’s Law Hence Duvall law Standing-wave condition, with surface phase shift a i.e. F(w) = (n+α)π/ω, where w = ω/L. Can determine RHS observationally and hence find F(w).

61 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, (c 2,  ) or (u, Y) where u=p/  and Y is the helium abundance.

62 Invoking hydrostatic equilibrium There appear to be three independent unknown functions: δp/p, δρ/ρ, and δΓ 1 /Γ 1. But the oscillations are presumed to take place about an equilibrium background in hydrostatic equilibrium: Perturbing this gives Likewise, using the mass equation, δm can be written in terms of δρ. Hence δp/p can finally be expressed in terms of δρ/ρ, and the number of unknown functions reduced from 3 to 2.

63 Structure Inversions Linearized 1d version, after taking difference from model values. Surface Term Error

64 Kernel for c 2 Kernel for ρ Kernels for sound speed and density