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

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2. Data analysis

2.1. Maximum likelihood estimation

We used a maximum likelihood method for fitting the peaks in the observed Fourier spectra. We derived the amplitudes, linewidths, frequencies and noise characteristics represented by [FORMULA]. Instead of fitting the power spectra ([FORMULA]) by minimizing the well-known expression given by Duvall & Harvey (1986), we have, in this work, fitted the Fourier spectra [FORMULA] using the equation given by Paper I, and adapted from Schou (1992).

The main difference between the two methods lies in the way they treat the spectrum correlation due to the mode leakage. The first method takes into account only the diagonal terms of the covariance matrix [FORMULA], assuming that the observed spectra are statistically independent from each other and have a [FORMULA] probability density function with 2 degrees of freedom. However, from Eq. (1), the observed spectra are a linear combination of the single modes, i.e. they are statistically dependent upon one another and have a multivariate normal probability density function. In this method, the leakage between the modes is taken into account in a more complete way (for a detailed discussion see our previous work: Paper I and II).

We have applied this fitting technique to 360 days of GONG data starting on 23 August 1995 with a 87[FORMULA] duty cycle. The GONG consists of a network of 6 sites with solar velocity imagers ([FORMULA] 250 [FORMULA] 250 pixels) located around the Earth to obtain nearly continuous observations of the Sun's five-minute oscillations (for a detailed description, see Harvey et al. 1996). After extracting a single ([FORMULA]) mode by spherical harmonic decomposition, an FFT of the whole time series obtained through the merging of all the sites data is computed. No sidebands are visible due to the high continuity in the data. Fig. 1 shows the power spectra of the time series for [FORMULA] and [FORMULA] modes.

[FIGURE] Fig. 1. Power spectra of [FORMULA], [FORMULA] multiplet. From bottom to top, the power spectrum of [FORMULA], 0 and [FORMULA] modes. The vertical line indicates the mode frequency for [FORMULA]. The rotational splitting can be clearly seen as a displacement of [FORMULA] mode towards higher frequency and of [FORMULA] towards lower frequencies. The contribution of [FORMULA] in the [FORMULA] spectrum is seen as a structure of peaks on the right side of the vertical line.

A Lorentzian profile was used as a model of the variance of the spectrum of a single mode ([FORMULA]), [FORMULA] or, in a different notation, [FORMULA]:

[EQUATION]

where [FORMULA] is the power amplitude, [FORMULA] the full linewidth at half maximum and [FORMULA] is the central frequency. These parameters [FORMULA] plus the background noise ([FORMULA]) are the ones that we want to find. We used the same linewidth for the entire multiplet (Appourchaux et al. 1995; Rabello-Soares et al. 1997). Also, we used the same amplitude for the [FORMULA] modes; it is expected that the prograde and retrogade modes have the same amplitude and, in fact, we made some tests where they proved to be very similar. The same background noise is fitted for the [FORMULA] modes with [FORMULA]. However, for higher degree modes, a simplification was made: the background noise was shown to be very similar for all [FORMULA]s in a multiplet and the convergence of the fitting is easier using only one parameter for the multiplet. We kept the [FORMULA] background noise parameters for low degree modes where the central frequency accuracy is more sensitive to the fitting.

The frequency of the mode ([FORMULA]) is given by:

[EQUATION]

where [FORMULA] is the mean frequency of the mode multiplet and [FORMULA] are orthogonal polynomials with degree i defined by: [FORMULA] and [FORMULA] for [FORMULA]. The polynomials [FORMULA] can be expressed in terms of the Clebsch-Gordan coefficients (Pijpers 1997). The [FORMULA] coefficients measure the frequency splitting due to the solar rotation and are called splitting coefficients . We fitted for [FORMULA], [FORMULA] ([FORMULA]); for [FORMULA] and 3, only [FORMULA], [FORMULA] and [FORMULA] ([FORMULA]); and for [FORMULA], [FORMULA]-[FORMULA] ([FORMULA]).

We have fitted around the frequency of a given ([FORMULA]) mode in a 17-µHz window for [FORMULA], and in a 22-µHz window for [FORMULA]. However, when the mode frequency was higher than 4000 µHz, a larger window (50 µHz) was used to cope with the larger mode linewidth. An estimate of the uncertainties on the fitted parameters was obtained from the inverse of the Hessian matrix (see Paper I).

2.2. Leakage matrix computation

An important point in the p-mode parameter determination is the correct calculation of the leakage matrix [FORMULA] (Eq. 1) in fitting the power or Fourier spectra. Following Schou & Brown (1994), the calculation of the leakage matrix [FORMULA] of the GONG data was described in Paper II.

The leakage matrix has some interesting properties. When [FORMULA] is odd, its elements are zero because of the parity of the spherical harmonics. The leakage between modes with different m's but the same l are such that: [FORMULA], [FORMULA] and [FORMULA], resulting in [FORMULA] elements to be calculated in a given multiplet. Note that, in this case, [FORMULA] is not symmetrical after applying the normalization [FORMULA]. In the following sections, only the leakage within a given multiplet will be considered. Only in Sect. 3.3 will the problem of leakage between different l's be addressed. Although the leakage matrix changes throughout the year due to the [FORMULA] angle variation, we assumed that over one year this variation averages out.

In Paper II, we developed and described a powerful test to check the correctness of the calculated leakage elements by looking directly in the observed Fourier spectra, namely the cross echelle diagram . It was applied to the GONG data with success, in particular for [FORMULA] and 2 modes.

2.3. Noise covariance matrix estimation

The noise covariance matrix [FORMULA] (Eq. 2) could be determined by an expression similar to the leakage matrix (see Schou & Brown 1994 and Paper I). In fact its elements, for which [FORMULA] is odd, are zero in the same way as for the leakage matrix elements (Papers I and II). Another way to calculate the noise covariance matrix, and one which will be used here, is to estimate it directly from the cross spectrum outside the p-mode band. Unfortunately, the different source of solar and instrumental noise introduces a frequency dependence in the noise covariance elements. It is thus not obvious which part of the frequency spectrum to use. As the p modes are equally spaced in frequency, the noise covariance is calculated in between the modes with radial order n and [FORMULA] to avoid a possible spurious frequency dependence. This is a tricky task because we have to be aware of the spurious modes present in the Fourier spectrum of the target mode, particularly, of modes with a different degree. However, the spherical harmonic decomposition is very efficient for modes with a degree very different from the target mode.

To calculate the noise covariance between the [FORMULA] Fourier spectra within a multiplet with a given [FORMULA], we selected windows of approximately 20 µHz width in between its central frequency [FORMULA] and the adjacent ones [FORMULA] and [FORMULA], carefully avoiding frequency regions where there are modes with a degree [FORMULA] where [FORMULA]. We used windows as large as possible to average out higher l modes which could be present in the spectra with a very small amplitude. In practice, these windows are better visualized in an echelle diagram centered on the target degree and containing all modes with [FORMULA] not very different from the target l.

Instead of estimating the noise covariance directly, we computed the ratio of the noise covariance matrix [FORMULA] because it varies slowly with frequency and we can, in most cases, fit a straight line to its frequency dependence:

[EQUATION]

where the average is over the 20 µHz windows defined above and selected for each radial order n. The average inside the selected window of the ratio cross spectrum within a multiplet with a given [FORMULA] is calculated and plotted against frequency (see Fig. 2 for an example). The frequency dependence introduced by different sources of solar and instrumental noise is larger in some of the noise covariance matrix elements than others. We find that a polynomial of degree 1 and, in some cases, of degree 3 are a good fitting to [FORMULA]. The theoretical calculation of the noise covariance matrix taking into account the solar and instrumental noise would be desirable to fully understand the [FORMULA] behaviour; however, this is outside the scope of this paper.

[FIGURE] Fig. 2. Measured averages of the ratio cross spectrum (circles) for [FORMULA]. On the top , [FORMULA] and [FORMULA]. On the bottom , [FORMULA] and [FORMULA]. Their frequency dependence is clearly seen. The continuous line is the polynomial fitting.

Note that [FORMULA] is the noise level of the observed power spectrum: [FORMULA]. Thus, the noise covariance matrix is constructed using the estimated ratio cross spectrum:

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
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