TY - JOUR

T1 - Fourier analysis of gapped time series

T2 - Improved estimates of solar and stellar oscillation parameters

AU - Stahn, Thorsten

AU - Gizon, Laurent

N1 - Funding Information:
The MLE source code is available from the Internet platform of the European Helio-and Asteroseismology Network (HELAS, funded by the European Union) at http://www.mps. mpg.de/projects/seismo/MLE_SoftwarePackage/.

PY - 2008/9

Y1 - 2008/9

N2 - 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.

AB - 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.

KW - Helioseismology: observations

KW - Oscillations: solar

KW - Oscillations: stellar

UR - http://www.scopus.com/inward/record.url?scp=50249105178&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50249105178&partnerID=8YFLogxK

U2 - 10.1007/s11207-008-9181-0

DO - 10.1007/s11207-008-9181-0

M3 - Article

AN - SCOPUS:50249105178

SN - 0038-0938

VL - 251

SP - 31

EP - 52

JO - Solar Physics

JF - Solar Physics

IS - 1-2

ER -