Abstract
Quantitative helioseismology and asteroseismology require very precise measurements of the frequencies, amplitudes, and lifetimes of the global modes of stellar oscillation. The precision of these measurements depends on the total length (T), quality, and completeness of the observations. Except in a few simple cases, the effect of gaps in the data on measurement precision is poorly understood, in particular in Fourier space where the convolution of the observable with the observation window introduces correlations between different frequencies. Here we describe and implement a rather general method to retrieve maximum likelihood estimates of the oscillation parameters, taking into account the proper statistics of the observations. Our fitting method applies in complex Fourier space and exploits the phase information. We consider both solar-like stochastic oscillations and long-lived harmonic oscillations, plus random noise. Using numerical simulations, we demonstrate the existence of cases for which our improved fitting method is less biased and has a greater precision than when the frequency correlations are ignored. This is especially true of low signal-to-noise solar-like oscillations. For example, we discuss a case where the precision of the mode frequency estimate is increased by a factor of five, for a duty cycle of 15%. In the case of long-lived sinusoidal oscillations, a proper treatment of the frequency correlations does not provide any significant improvement; nevertheless, we confirm that the mode frequency can be measured from gapped data with a much better precision than the 1/T Rayleigh resolution.
Helioseismology, Asteroseismology, and MHD Connections
Guest Editors: Laurent Gizon and Paul Cally
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Anderson, E.R., Duvall, T.L., Jefferies, S.M.: 1990, Astrophys. J. 364, 699.
Appourchaux, T., Gizon, L., Rabello-Soares, M.-C.: 1998, Astron. Astrophys. Suppl. Ser. 132, 107.
Appourchaux, T., Chang, H.-Y., Gough, D.O., Sekii, T.: 2000, Mon. Not. Roy. Astron. Soc. 319, 365.
Bedding, T.R., Kjeldsen, H.: 2007, Commun. Asteroseismol. 150, 106.
Brandt, S.: 1970, Statistical and Computational Methods in Data Analysis, North-Holland, Amsterdam.
Butler, R.P., Bedding, T.R., Kjeldsen, H., McCarthy, C., O’Toole, S.J., Tinney, C.G., Marcy, G.W., Wright, J.T.: 2004, Astrophys. J. 600, L75.
Cuypers, J.: 1987, Bull. Acad. Roy. Sci. Belg. (Cl. Sci.) 49, 21.
Duvall, T.L., Harvey, J.W.: 1986, In: Gough, D.O. (ed.) NATO Advanced Research Workshop, Reidel, Dordrecht, 105.
Duvall, T.L., Jefferies, S.M., Harvey, J.W., Osaki, Y., Pomerantz, M.A.: 1993, Astrophys. J. 410, 829.
Fossat, E., Kholikov, S., Gelly, B., Schmider, F.X., Fierry-Fraillon, D., Grec, G., Palle, P., Cacciani, A., Ehgamberdiev, S., Hoeksema, J.T., Lazrek, M.: 1999, Astron. Astrophys. 343, 608.
Gabriel, M.: 1994, Astron. Astrophys. 287, 685.
Gizon, L., Solanki, S.K.: 2003, Astrophys. J. 589, 1009.
Goldreich, P., Keeley, D.A.: 1977, Astrophys. J. 212, 243.
Horn, R.A., Johnson, C.R.: 1985, Matrix Analysis, Cambridge University Press, Cambridge.
Kendall, M.G., Stuart, A.: 1967, The Advanced Theory of Statistics: Inference and Relationship, vol. 2, 2nd edn., Butler and Tanner, London.
Libbrecht, K.G.: 1992, Astrophys. J. 387, 712.
Miller, B.A., Hale, S.J., Elsworth, Y., Chaplin, W.J., Isaak, G.R., New, R.: 2004, In: Danesy, D. (ed.) Proc. SOHO 14/GONG 2004 Workshop: Helio- and Asteroseismology: Towards a Golden Future, SP-559, ESA, Noordwijk, 571.
Schou, J.: 1992, Ph.D. Dissertation, University of Aarhus.
Stein, R., Georgobiani, D., Trampedach, R., Ludwig, H.-G., Nordlund, Å, 2004, Solar Phys. 220, 229.
Toutain, T., Appourchaux, T.: 1994, Astron. Astrophys. 289, 649.
Winget, D.E., Nather, R.E., Clemens, J.C., Provencal, J., Kleinman, S.J., Bradley, P.A., Wood, M.A., Claver, C.F., Frueh, M.L., Grauer, A.D., Hine, B.P., Hansen, C.J., Fontaine, G., Achilleos, N., Wickramasinghe, D.T., Marar, T.M.K., Seetha, S., Ashoka, B.N., O’Donoghue, D., Warner, B., Kurtz, D.W., Buckley, D.A., Brickhill, J., Vauclair, G., Dolez, N., Chevreton, M., Barstow, M.A., Solheim, J.E., Kanaan, A., Kepler, S.O., Henry, G.W., Kawaler, S.D.: 1991, Astrophys. J. 378, 326.
Woodard, M.F.: 1984, Ph.D. Dissertation, University of California, San Diego.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science+Business Media, BV
About this chapter
Cite this chapter
Stahn, T., Gizon, L. (2008). Fourier Analysis of Gapped Time Series: Improved Estimates of Solar and Stellar Oscillation Parameters. In: Gizon, L., Cally, P., Leibacher, J. (eds) Helioseismology, Asteroseismology, and MHD Connections. Springer, New York, NY. https://doi.org/10.1007/978-0-387-89482-9_4
Download citation
DOI: https://doi.org/10.1007/978-0-387-89482-9_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-89481-2
Online ISBN: 978-0-387-89482-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)