Indirect imaging of stellar non-radial pulsations

Svetlana V. Berdyugina

University of Oulu, Finland

Institute of Astronomy, ETH Zurich, Switzerland

Moletai, August 2005 2

Overview

Inversion methods in astrophysics Inverse problem Maximum likelihood method Regularization

Stellar surface imaging Line profile distortions Localization of inhomogeneities

Imaging of stellar non-radial pulsations Temperature variations Velocity field

Mode identification sectoral modes: symmetric tesseral modes: antisymmetric tesseral modes: zonal modes:

||ml 2|| ml

1|| ml0m

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1. Inversion methods in astrophysics

Inverse problem

Maximum likelihood method

Regularization Maximum Entropy Tikhonov Spherical harmonics Occamian approach

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Inverse problem

Determine true properties

of phenomena (objects)

from observed effects

All problems in astronomy

are inverse

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Inverse problem

Trial-and-error method Response operator (PSF, model) is known Direct modeling while assuming various properties of the object

Inversion True inversion: unstable solution due to noise ill-posed problem Parameter estimation: fighting the noise

Data ObjectResponse operator

PSD

DPS 1

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Inverse problem

Estimate true properties

of phenomena (objects)

from observed effects

Parameter estimation

problem

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Maximum likelihood method

Probability density function (PDF):

Normal distribution:

Likelihood function

Maximum likelihood

)( SDf

2

2

2

)]([

2

1)(

SRD

eSDf

m

jj SDfSL

1

)()(

max)(lnmax)( SLSL

)(SR

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Maximum likelihood method

Maximum likelihood

Normal distribution

Residual minimization

niSLSi

,1,0)(ln

max)]([

1

2

m

j

jj

m

SRD

min)]([

1

2

m

j

jj

m

SRD

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Maximum likelihood method

Maximum likelihood solution: Unique Unbiased Minimum variance UNSTABLE !!!

Reduce the overall probability

Statistical tests test Kolmogorov Mean information

0LL

2

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Maximum likelihood method

A multitude of solutions with probability

New solution Biased only within noise level Stable NOT UNIQUE !!!

Lik

elih

oo

d

Solutions

0LL

0L

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Regularization

Provide a unique solution Invoke additional constraints Assign special properties of a new solution

Maximize the functional

max)()(ln SgSL

Regularized solution is forced

to possess properties

)(Sg

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Bayesian approach

Thomas Bayes (1702-1761) Posterior and prior probabilities

Prior information on the solution

Using a priori constraints

is

the Bayesian approach

)(Sg

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Maximum entropy regularization

Entropy In physics: a measure of ”disorder” In math (Shannon): a measure of “uninformativeness”

Maximum entropy method (MEM, Skilling & Bryan, 1984):

MEM solution Largest entropy (within the noise level of data) Minimum information (minimum correlation)

s

sPsPSH )(log)()(

SSSg log)(

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Tikhonov regularization

Tikhonov (1963):

Goncharsky et al. (1982):

TR solution Least gradient (within the noise level of data) Smoothest solution (maximum correlation)

2||||)( SSg

2||||)( SSg

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Spherical harmonics regularization

Piskunov & Kochukhov (2002): multipole regularization

MPR solution Closest to the spherical harmonics expansion Can be justified by the physics of a phenomenon

Mixed regularization:

2||||)( mpolSSSg

ml

mllm

mpol YaS,

),(

22

21 |||||||| mpolSSS

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Occamian approach

William of Occam (1285 --1347): Occam's Razor: the simplest explanation

to any problem is the best explanation

Terebizh & Biryukov (1994, 1995): Simplest solution (within the noise level of data) No a priori information

Fisher information matrix:

PDPSF mT ][)( ,,1

PSD

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Occamian approach

Orthogonal transform

Principal components

Simplest solution

Unique Stable

VVSF T)(

SVSY T)(

)()( pp VYS

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Key issues

Inverse problem is to estimate true properties of phenomena (objects) from observed effects

Maximum likelihood method results in the unique but unstable solution

Statistical tests provide a multitude of stable solutions

Regularization is needed to choose a unique solution

Regularized solution is forced to possess assigned properties

MEM solution minimum correlation between parameters

TR solution maximum correlation between parameters

MPR solution closest to the spherical harmonics expansion

OA solution simplest among statistically acceptable