Indirect imaging of stellar nonradial pulsations Svetlana V

Indirect imaging of stellar non-radial pulsations Svetlana V. Berdyugina University of Oulu, Finland Institute of Astronomy, ETH Zurich, Switzerland Moletai, August 2005

Overview Inversion methods in astrophysics Ø Inverse problem Ø Maximum likelihood method Ø Regularization Stellar surface imaging Ø Line profile distortions Ø Localization of inhomogeneities Imaging of stellar non-radial pulsations Ø Temperature variations Ø Velocity field Mode identification Ø Ø sectoral modes: symmetric tesseral modes: antisymmetric tesseral modes: zonal modes: Moletai, August 2005 2

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

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

Inverse problem Data Response operator Object Trial-and-error method Ø Response operator (PSF, model) is known Ø Direct modeling while assuming various properties of the object Inversion Ø True inversion: unstable solution due to noise Ø ill-posed problem Ø Parameter estimation: fighting the noise Moletai, August 2005 5

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

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

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

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

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

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

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

Maximum entropy regularization Entropy Ø In physics: a measure of ”disorder” Ø In math (Shannon): a measure of “uninformativeness” Maximum entropy method (MEM, Skilling & Bryan, 1984): MEM solution Ø Largest entropy (within the noise level of data) Ø Minimum information (minimum correlation) Moletai, August 2005 13

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

Spherical harmonics regularization Piskunov & Kochukhov (2002): multipole regularization MPR solution Ø Closest to the spherical harmonics expansion Ø Can be justified by the physics of a phenomenon Mixed regularization: Moletai, August 2005 15

Occamian approach William of Occam (1285 --1347): Ø Occam's Razor: the simplest explanation to any problem is the best explanation Terebizh & Biryukov (1994, 1995): Ø Simplest solution (within the noise level of data) Ø No a priori information Fisher information matrix: Moletai, August 2005 16

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

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 18