Up: The art of fitting
In the past decade, helioseismology has been able to provide the internal
structure of the Sun and its dynamics. These inferences have been made
possible by inverting the frequencies and rotational splitting of the
pressure modes. The most commonly used technique for obtaining the p-mode
parameters is to fit the p-mode spectra using Maximum Likelihood Estimators
(MLE) assuming that the statistical distribution of the p modes in the
power spectra is a with 2 degrees of freedom (Woodard
1984). The MLE with this statistics were first applied on
helioseismology data by Duvall & Harvey (1986) and
Anderson et al. (1990). This technique is used for fitting
spectra obtained with integrated sunlight instruments. For low- or
high-resolution instruments, the power spectra are commonly fitted
assuming that each m spectrum has the same statistics as the for the
integrated sunlight instruments (LOI
instrument: Appourchaux et al. 1995; Rabello-Soares et al.
1997; GONG instrument: Hill et al. 1996).
of these implementations are correct since the assumed statistics is wrong.
Only Schou (1992) described a more correct way of fitting
diagrammes using not the power spectra
but the complex Fourier spectra.
The pioneering work of Schou (1992) has inspired this series of
3 articles for addressing our state of the art of fitting
diagrams. In this paper (Part I), we describe the statistics of the p modes, and
how the MLE can be used in helioseismology. In Appourchaux et al.
(hereafter Part II), we
show how one can measure the mode leakage matrix and the noise
correlation from the data which knowledge is required for using the Part
I. In Appourchaux & Gizon (1998) (hereafter Part III), we
will apply these techniques to the LOI instrument of VIRGO on board SOHO
(For a description of the performance of the instrument see Appourchaux
et al. 1997).
In this paper, we explain how the MLE can be used in helioseismology. In the first section, we recall the properties of MLE.
In the second
section, we describe the statistics of the p-mode Fourier spectra. In
this section, we have generalized the approach of Schou (1992),
to any complex leakage matrices. We have also used complex matrices to
generate the covariance matrices of the p modes and of the noise. In the
third section we show how to use Monte-Carlo simulations for testing
both the use of MLE and the model of the p-mode spectra, and then
Up: The art of fitting
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