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Astron. Astrophys. 345, 1027-1037 (1999)

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1. Introduction

Helioseismology 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 l) penetrates deeper into the solar interior than modes with a smaller wavelength. These low-degree modes give us information on the solar core. However, even if our aim is to study only the solar core, the low-degree modes must be combined with high-degree modes because the dominant contributions to the mode eigenfunction come from the outer parts of the Sun. To detect modes with [FORMULA], it is necessary to have observations with spatial resolution.

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 ([FORMULA]) 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 ([FORMULA]) mode ([FORMULA]) is not given by the Fourier transform of a single mode ([FORMULA]), but rather as a sum over several modes, which can be expressed as:

[EQUATION]

where the sum comprises the modes contributing to the given spectrum and [FORMULA] is the contribution from each mode. [FORMULA] is called the leakage matrix: it is a square matrix with each dimension equal to [FORMULA], where again the sum comprises the modes contributing to the given spectrum [FORMULA]. The [FORMULA] elements are given by [FORMULA] where a given [FORMULA] characterizes a column and a given [FORMULA] a line. [FORMULA] is independent of frequency [FORMULA] 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 [FORMULA], which is in general represented by a Lorentzian shape with a set of parameters [FORMULA]. For obvious reasons, the leakage matrix is normalized as [FORMULA]. The expected covariance matrix of the real (or the imaginary) part of observed spectra ([FORMULA]) is given by:

[EQUATION]

where [FORMULA] is the variance of the real (or imaginary) part of the Fourier spectrum of a single mode ([FORMULA]) and [FORMULA] is the noise covariance matrix (Paper I). The power spectra give an estimate of the diagonal elements of [FORMULA] (Appourchaux et al. 1998b, hereafter Paper II). [FORMULA] and [FORMULA] have the same dimension as [FORMULA], and their elements are given by [FORMULA] and [FORMULA] 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 ([FORMULA]) 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.

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
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