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Indirect imaging of stellar non-radial pulsations

Indirect imaging of stellar non-radial pulsations. Svetlana V. Berdyugina University of Oulu, Finland Institute of Astronomy, ETH Zurich, Switzerland. Overview. Inversion methods in astrophysics Inverse problem Maximum likelihood method Regularization Stellar surface imaging

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Indirect imaging of stellar non-radial pulsations

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  1. Indirect imaging of stellar non-radial pulsations Svetlana V. Berdyugina University of Oulu, Finland Institute of Astronomy, ETH Zurich, Switzerland

  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: Moletai, August 2005

  3. 1. Inversion methods in astrophysics • Inverse problem • Maximum likelihood method • Regularization • Maximum Entropy • Tikhonov • Spherical harmonics • Occamian approach Moletai, August 2005

  4. All problems in astronomy are inverse Inverse problem Determine true properties of phenomena (objects) from observed effects Moletai, August 2005

  5. Response operator Data Object 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 Moletai, August 2005

  6. Parameter estimation problem Inverse problem Estimate true properties of phenomena (objects) from observed effects Moletai, August 2005

  7. Maximum likelihood method • Probability density function (PDF): • Normal distribution: • Likelihood function • Maximum likelihood Moletai, August 2005

  8. Maximum likelihood method • Maximum likelihood • Normal distribution • Residual minimization Moletai, August 2005

  9. Maximum likelihood method • Maximum likelihood solution: • Unique • Unbiased • Minimum variance • UNSTABLE !!! • Reduce the overall probability • Statistical tests • test • Kolmogorov • Mean information Moletai, August 2005

  10. Likelihood Solutions Maximum likelihood method • A multitude of solutions with probability • New solution • Biased only within noise level • Stable • NOT UNIQUE !!! Moletai, August 2005

  11. Regularization • Provide a unique solution • Invoke additional constraints • Assign special properties of a new solution • Maximize the functional Regularized solution is forced to possess properties Moletai, August 2005

  12. Bayesian approach Using a priori constraints is the Bayesian approach • Thomas Bayes (1702-1761) • Posterior and prior probabilities • Prior information on the solution Moletai, August 2005

  13. 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) Moletai, August 2005

  14. Tikhonov regularization • Tikhonov (1963): • Goncharsky et al. (1982): • TR solution • Least gradient (within the noise level of data) • Smoothest solution (maximum correlation) Moletai, August 2005

  15. 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: Moletai, August 2005

  16. 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: Moletai, August 2005

  17. Occamian approach • Orthogonal transform • Principal components • Simplest solution • Unique • Stable Moletai, August 2005

  18. 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 Moletai, August 2005

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