## 1. IntroductionHelioseismology uses the temporal and spatial properties of solar
oscillations to study the solar interior. Acoustic modes with a large
horizontal wavelength (corresponding to a low spherical harmonic
degree When the solar image is resolved a very large number of modes can be detected making the data analysis very complicated. In order to extract a single () mode from resolved observations, spherical harmonic decomposition of each image must be performed. Unfortunately, it is essentially impossible to isolate individual modes based on their angular dependence alone, because only half of the solar surface is observed. Since the modes are not completely isolated, the Fourier transform for a given () mode () is not given by the Fourier transform of a single mode (), but rather as a sum over several modes, which can be expressed as: where the sum comprises the modes contributing to the given spectrum and is the contribution from each mode. is called the leakage matrix: it is a square matrix with each dimension equal to , where again the sum comprises the modes contributing to the given spectrum . The elements are given by where a given characterizes a column and a given a line. is independent of frequency and it is the same for both real and imaginary parts of the Fourier transform (Schou 1992; Appourchaux et al. 1998a, hereafter Paper I). The Fourier spectra of the p modes are normally distributed with a zero mean and variance , which is in general represented by a Lorentzian shape with a set of parameters . For obvious reasons, the leakage matrix is normalized as . The expected covariance matrix of the real (or the imaginary) part of observed spectra () is given by: where is the variance of the real (or imaginary) part of the Fourier spectrum of a single mode () and is the noise covariance matrix (Paper I). The power spectra give an estimate of the diagonal elements of (Appourchaux et al. 1998b, hereafter Paper II). and have the same dimension as , and their elements are given by and respectively. This is a rather important point to be considered in fitting the data. Not taking into account or considering in an incorrect manner, the spurious modes on a given () mode spectrum will produce an erroneous determination of the p-mode parameters, as already discussed by Appourchaux et al. (1995) and Rabello-Soares et al. (1997). This work presents the results of a very careful analysis of the low degree solar p modes observed by the Global Oscillation Network Group (GONG) with a careful treatment of the spatial leakage of the modes. © European Southern Observatory (ESO) 1999 Online publication: April 28, 1999 |