Spectrum Estimation in Helioseismology

P.B. Stark

Department of StatisticsUniversity of CaliforniaBerkeley CA 94720-3860 www.stat.berkeley.edu/~stark

Source: GONG website

Acknowledgements• Most media pirated from websites of

– Global Oscillations Network Group (GONG)

– Solar and Heliospheric Observer (SOHO) Solar Oscillations Investigation

• Collaboration w/ S. Evans (UCB), I.K. Fodor (LLNL), C.R. Genovese (CMU), D.O.Gough (Cambridge), Y. Gu (GONG), R. Komm (GONG), M.J. Thompson (QMW)Source: sohowww.nascom.nasa.gov

Source: www.gong.noao.edu

The Difference between Theory and PracticeIn Theory, there is no difference between Theory and Practice, but in Practice, there is.I’m embarrassed to give a talk about practice to this lofty audience of mathematicians.My first work in helioseismology was theory/methodology for inverse problems.Surprise! Data not K + , {i} iid, zero mean, Var(i) known—but everyone pretends so.

The Sun• Closest star, ~1.5*108 km• Radius ~6.96*105 km• Mass ~1.989*1030 kg• Teff ~5780 K• Luminosity ~3.846*1026 J/s• Surface gravity ~274 m/s2

• Age ~4.6*109y• Mean density ~1408 kg/m3

• Z/X ~0.02; Y ~0.24

The Sun Vibrates• Stellar oscillations known

since late 1700s. • Sun's oscillation observed in

1960 by Leighton, Noyes, Simon.

• Explained as trapped acoustic waves by Ulrich, Leibacher, Stein, 1970-1.

Source: SOHO-SOI/MDI website

Pattern is Superposition of Modes• Like vibrations of a

spherical guitar string• 3 “quantum numbers” l, m,

n• l and m are spherical

surface wavenumbers• n is radial wavenumber

Source: GONG website

Waves Trapped in Waveguide• Low l modes sample more

deeply• p-modes do not sample

core well• Sun essentially opaque to

EM; transparent to sound & to neutrinos

Source: forgotten!

Spectrum is very Regular• Explanation as modes, plus

stellar evolutionary theory, predict details of spectrum

• Details confirmed in data by Deubner, 1975

Source: GONG

More data for Sun than for Earth• Over 107 modes predicted• Over 250,000 identified• Will be over 106 soon

Formal error bars inflated by 200.Hill et al., 1996. Science 272, 1292-1295

“5-minute” oscillations• Takes a few hours for

energy to travel through the Sun.

• p-mode amplitude ~1cm/s• Brightness variation ~10-7

• Last from hours to months• Excited by convection

40-day time series of mode coefficients, speeded-up by 42,000. l=1, n=20; l=0, 1, 2, 3A. Kosovichev, SOHO website.

1 mode 3 modes all modes

Oscillations Taste Solar Interior• Frequencies sensitive to material properties• Frequencies sensitive to differential rotation• If Sun were spherically symmetric and did not rotate,

frequencies of the 2l+1 modes with the same l and n would be equal

• Asphericity and rotation break the degeneracy (Scheiner measured 27d equatorial rotation from sunspots by 1630. Polar ~33d.)

• Like ultrasound for the Sun

Different Modes sample Sun differently

l=20 modes. Left: m=20. Middle: m=16. (Doppler velocities)Right: section through eigenfunction of l=20, m=16, n =14.

Left: raypath for l=100, n=8 and l=2, n=8 p-modesRight: raypath for l=5, n=10 g-mode. g-modes have not been observed

Gough et al., 1996. Science 272, 1281-1283

Can combine Modes to target locally

Cuts through kernels for rotation:A: l=15, m=8. B: l=28, m=14. C: l=28, m=24.D: two targeted combinations: 0.7R, 60o; 0.82R, 30o

Thompson et al., 1996. Science 272, 1300-1305.

Estimated rotation rate as a function of depthat three latitudes.Source: SOHO-SOI/MDI website

Goals of Helioseismology

• Learn about composition, state, dynamics of closest star: sunspots, heliodynamo, solar cycle

• Test/improve theories of stellar evolution• Use Sun as physics lab: conditions unattainable on

Earth (neutrino problem, equation of state)• Predict space weather?

Successes so far

• Revised depth of the solar convection zone• Ruled out dynamo models with rotation constant on

cylinders • Found errors in opacity calculations of numerical

nuclear physicists • Progress in solar neutrino problem

Plasma Rivers in the Sun

• SOHO “imaged” rivers of solar plasma moving ~10% faster than the surrounding material.

• NASA top-10 story, 1997.

Source: SOHO SOI/MDI website

Sun Quakes• Can see acoustic waves

propagating from a solar flare

• Time-distance helioseismology: new field

Source: SOHO-SOI/MDI website

Current Experiments

Global Oscillations Network Group (GONG)

• 6-station terrestrial network; Sun never sets

• Funded by NSF

Solar and Heliospheric Observer Solar Oscillations Investigation (SOHO-SOI/MDI) satellite

• Orbits L1; Sun never sets• Funded by NASA and ESA

Experiments range from high-resolution to Sun-as-a-star.The most extensive with highest duty cycle are

Velocity from Doppler Shifts: Michelson Doppler Interferometer

• Measure amplitudes of 3 light frequencies in Ni I absorption band, 676.8nm

• Probes mid-photosphere• Get velocity in each pixel of

CCD image• Developed by T. Brown

(HAO/NCAR) in 1980'sSource: GONG website

Data Reduction

Harvey et al., 1996. Science 272, 1284-1286

Why Reduce the Data to Normal Modes?• GONG+ data rate ~4GB/day: can’t invert a year of data

Mode parameters are a much smaller set.(N.b. 4GB/day asymptopia)

• Identifying mode parameters helps separate the stochastic disturbance from the characteristics of the oscillator

• Relationship between the time-domain surface motion of a stochastically excited gaseous ball with magnetic stiffening not well understood

GONG Data Pipeline1. Read tapes from sites.2. Correct for CCD characteristics3. Transform intensities to Doppler velocities4. Calibrate velocities using daily calibration images5. Find image geometry and modulation transfer

function (atmospheric effects, lens dirt, instrument characteristics, ...)

6. High-pass filter to remove steady flows7. Remap images to heliographic coordinates,

interpolate, resample, correct for line-of-sight8. Transform to spherical harmonics: window,

Legendre stack in latitude, FFT in longitude

9. Adjust spherical harmonic coefficients for estimated modulation transfer function

10. Merge time series of spherical harmonic coefficients from different stations; weight for relative uncertainties

11. Fill data gaps of up to 30 minutes by ARMA modeling

12. Compute periodogram of time series of spherical harmonic coefficients

13. Fit parametric model to power spectrum by iterative approximate maximum likelihood

14. Identify quantum numbers; report frequencies, linewidths, background power, and uncertainties

Steps in Data Processing

Raw intensity image

Doppler velocity image

High-pass filtered to remove rotation & flows

And more steps…

Time series of spherical harmonic coefficients

Spectra of time series, and fitted parametric models

Top: GONG website. Bottom: Hill et al., Science, 1996.

Duty Cycle

• Both GONG and SOHO-SOI/MDI try to get uninterrupted data

• Other experiments at South Pole—long day• Gaps in data make it harder to estimate the

oscillation spectrum: artifacts in periodogram

Effect of Gaps• Don’t observe process of interest.• Observe process × window• Fourier transform of data is FT of

process, convolved with FT of window.• FT of window has many large sidelobes• Convolution causes energy to “leak”

from distant frequencies into any particular band of interest.

Power spectrum of window

95% duty cycle window

Tapering

• Want simplicity of periodogram, but less leakage• Traditional approach—multiply data by a smooth

“taper” that vanishes where there are no data• Smoother tapersmaller sidelobes, but more local

smearing (loss of resolution)• Pose choosing taper as optimization problem

• What taper minimizes “leakage” while maximizing resolution?

• Leakage is a bias; optimality depends on definition• Broad-band & asymptotic yield eigenvalue problems:

Optimal Tapering

2

2

maxargsupp:

w

Aw

wnw

taperoptimal

Prolate Spheroidal Tapers

• Maximize the fraction of energy in a band [-w, w] around zero

• Analytic solution when no gaps:– 2wT tapers nearly perfect– others very poor

• Must choose w • Character different with gaps

T = 1024, w = 0.004. Fodor&Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483

Minimum Asymptotic Bias Tapers

• Minimize integral of spectrum against frequency squared

• Leading term in asymptotic bias

T = 1024 Fodor&Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483

Sine Tapers

• Without gaps, approximate minimum asymptotic bias tapers

• With gaps, reorthogonalize w.r.t. gap structure

T = 1024.Fodor&Stark, 2000. IEEE Trans. Sig. Proc. 48, 3472-3483.

Optimization Problems

• Prolate and minimum asymptotic bias tapers are top eigenfunctions of large eigenvalue problems

• The problems have special structure; can be solved efficiently (top 6 tapers for T=103,680 in < 1day)

• Sine tapers very cheap to compute

Sample Concentration of TapersT=1024, w = 0.004

Order Prolate Prolate w/ gaps Projected prolate

0 0.9999 0.9980 0.81261 0.9999 0.9977 0.97042 0.9999 0.9785 0.93173 0.9999 0.9753 0.97224 0.9999 0.9609 0.9550

Fodor & Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483

Sample Asymptotic Bias of Tapers, T = 1024Order Minimum Asympt Bias Min Bias w/ gaps Projected Min Bias

0 0.00021 0.00437 0.735161 0.00090 0.02265 0.431732 0.00212 0.13854 0.638843 0.00382 0.23002 0.559214 0.00594 0.27385 0.45409

Fodor & Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483

Multitaper Estimation

• Top several eigenfunctions have eigenvalues close to 1.• Eigenvalues drop to zero, abruptly for no-gap• Estimates using orthogonal tapers are asymptotically

independent (mild conditions)• Averaging spectrum estimates from several “good” tapers can

decrease variance without increasing bias much.• Get rank K quadratic estimator.

Multitaper Procedure

• Compute K orthogonal tapers, each with good concentration

• Multiply data by each taper in turn• Compute periodogram of each product• Average the periodogramsSpecial case: break data into segments

Cheapest is Fine

For simulated and real helioseismic time series of length T=103,680, no discernable systematic difference among 12-taper multitaper estimates using the three families of tapers.

Use re-orthogonalized sine tapers because they are much cheaper to compute, for each gap pattern in each time series.

Multitaper Simulation• Can combine with

segmenting to decrease dependence

• Prettier than periodogram; less leakage.

• Better, too?T=103,680. Truth in grey. Left panels: periodogram. Right panels: 3-segment 4-taper gapped sine taper estimate. Fodor&Stark, 2000.

Multitaper: SOHO Data• Easier to identify mode

parameters from multitaper spectrum

• Maximum likelihood more stable; can identify 20% to 60% more modes (Komm et al., 1999. Ap.J., 519, 407-421) SOHO l=85, m=0. T = 103, 680.

Periodogram (left) and 3-segment 4 sine taper estimateFodor&Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483

Error bars: Confidence Level in SimulationMethod % Coverage Method % Coverage

Parametric chi-square 72 Blockwise bootstrap 56

Simulation from Estimate 82 Bootstrap pivot percentile 71

Jackknife 75 Pre-pivot bootstrap 78

Bootstrap normal 77 Iterated pre-pivot bootstrap 91

Bootstrap-t 78 Bootstrap percentile of percentiles 90

Bootstrap percentile 69 Bootstrap pivot percentile of percentiles 96

1,000 realizations of simulated normal mode data. 95% target confidence level. Fodor & Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483

Pivot• Asymptotic pivot: asymptotic distribution does not

depend on unknown parameter• For K multitaper estimate, = 10log10S(f),

)(2

2(0.4343)10σ,0θ2

KN

DT

Pivot: Rn= (T- )/

Depth of Convection Zone• D. O. Gough showed using helioseismic data

that the solar convection zone probably is rather thicker than had been thought.

Falsified dynamo models

Since the mid-1980s, many studies of solar rotation using frequency splittings (broken degeneracy from rotation) have shed doubt on dynamo models that required rotation to be roughly constant on cylinders in the convection zone.

Errors in Opacity Calculations• The “standard solar model” fit the estimates of soundspeed better

with opacity at base of convection zone modified in an ad hoc way. • Physicists who calculated original opacity found bound-state

contribution of iron had been underestimated.• Led to 10-20% error in opacity at base of convection zone.• Revised opacities fit solar data• Explained mysterious pulsation period ratios of Cepheid stars

Duval, 1984. Nature, 310, p22.

Solar Neutrino Problem• Measurements of solar neutrino flux lower than

predicted by nuclear physics + stellar evolution models• Not clear whether problem with nuclear physics or

with theory of stellar evolution• Helioseismology suggests low Helium abundance in

core not plausible explanation

Linear Inverse Problem for Rotation

dθdθ),(κν rrrlmnlmn

Assumes eigenfunctions and radial structure known.

Consistency in Linear Inverse Problems• Xi = i + i, i=1, 2, 3, …

subset of separable Banach{i} * linear, bounded on {i} iid

consistently estimable w.r.t. weak topology iff{Tk}, Tk Borel function of X1, . . . , Xk s.t. , >0, *, limk P{|Tk - |>} = 0

• µ a prob. measure on ; µa(B) = µ(B-a), a • Hellinger distance• Pseudo-metric on **:

• If restriction to converges uniformly on increasing sequence of compacts to metric compatible with weak topology, and those compacts are totally bdd wrt d, can estimate consistently in weak topology.

• For given sequence of functionals {i}, µ rougher consistent estimation easier.

Importance of Error Distribution

kkii

k

ikk Ctt

CttD lim)κ,κ(δ1),( ,}{ 2/1

211

221

2/12ba2

1 }{ )( dμdμ b)δ(a,

Normal is Hardest

Suppose = {: [0,1], T Hilbert}• If {i} iid U[0,1], consistent iff |<i, >|=.

• If {i} iid N[0,1], consistent iff |<i, >|2 =