1 Introduction

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 $\chi^{2}$ 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 $(m,\nu)$ 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). Unfortunately, none of these implementations are correct since the assumed statistics is wrong. Only Schou (1992) described a more correct way of fitting $(m,\nu)$ 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 $(m,\nu)$ 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.  (1997) (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 conclude.