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

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3. Source of systematic errors

3.1. Fourier versus power spectra fitting

In Paper I, using Monte-Carlo simulation, we showed that there is no bias on the p-mode frequency determination using the power or Fourier fitting, while the splitting coefficients derived fitting the power spectra for [FORMULA] are slightly overestimated with a bias of about 10 nHz. Besides, the uncertainties of the fitted splitting coefficients for [FORMULA] modes are larger when fitting the power spectra and similar for [FORMULA] (see Fig. 8 in Paper I). These predict quite well the differences between the two techniques when applied to the observations presented here.

For [FORMULA], the splitting coefficients found fitting the power spectra present a larger range of values and their weighted average over n is 16 nHz higher than the Fourier fitting (see Fig. 3). As expected, the differences between the two approaches diminish for higher degree modes. In fact, for [FORMULA], the weighted average over n of the [FORMULA] splitting coefficient is higher by only 3 nHz fitting the power spectra and for [FORMULA], they agree quite well.

[FIGURE] Fig. 3. Absolute difference between the [FORMULA] coefficients and its weighted mean over n for [FORMULA] fitting the power spectra (stars) and the Fourier spectra (full circles). They are slightly displaced in frequency for a better visualization. The weighted average values fitting the power and the Fourier spectra are, respectively, [FORMULA] nHz and [FORMULA] nHz (in sidereal units).

We can conclude that the Fourier fitting is very important in determining [FORMULA] mode parameters and still important for [FORMULA] modes because it calculates all their parameters with the smallest bias and error bars. On the other hand, and despite the fact that no improvement was found for [FORMULA] modes fitting the Fourier spectra, which will probably be true for higher degree modes, it is still advisable to fit the Fourier spectra instead of the power spectra. Fitting Fourier spectra will considerably reduce the bias on the splitting coefficients.

3.2. Influence of the leakage matrix on the splitting coefficients determination

The correct calculation of the leakage matrix is particularly important in the determination of the splitting coefficients. A change in the values assumed for the leakage matrix elements will introduce systematic effects on the estimated splitting coefficients fitting the Fourier or the power spectra. This could be one of the reasons for the discrepancies between the measurements of the splitting coefficients by different instruments with spatial resolution. However, the central frequencies will not be affected by these changes (see also Paper I).

In Paper I, we estimated mode parameters for [FORMULA] and 2 by fitting the Fourier spectra of synthetic data using Monte Carlo simulation. We used values of the leakage matrix elements different from the known true ones and the systematic error made on the fitted splitting coefficients proved to be quadratic, which is a very interesting result.

Due to symmetry properties, for [FORMULA], there will be only one matrix element to be determined, i.e. the leakage between [FORMULA] and [FORMULA], which we called [FORMULA]. We undertook the same exercise here, using values of [FORMULA] different from that of the theoretical value [FORMULA] (Sect. 2.2), the [FORMULA] splitting coefficient was determined by fitting both the Fourier spectra and the power spectra (Fig. 4), obtaining results for the Fourier fitting with the same order of variation as found in the simulations. While the [FORMULA] coefficients calculated using the power spectra vary almost linearly with [FORMULA], those found using the Fourier spectra obey a quadratic law. Thus, when fitting the Fourier spectra, the [FORMULA] coefficient could only be underestimated using an incorrect value for [FORMULA]; its correct value corresponds to the maximum value of the [FORMULA] coefficient. Although these variations in the splitting coefficient are small in comparison to their error bars (around 15 nHz for an error of 20[FORMULA] in [FORMULA]), they are significant because of their systematic nature.

[FIGURE] Fig. 4. Weighted average over n of sidereal [FORMULA] for [FORMULA] as a function of the relative leakage [FORMULA] ([FORMULA]), where [FORMULA]. The full circles are found fitting the Fourier spectra and the triangles fitting the power spectra. The continuous line is the non-linear squares fit of the theoretical calculation (Eq. 7) to the [FORMULA] weighted average (full circles).

We have analytically derived the dependence of [FORMULA] upon [FORMULA] fitting the Fourier spectra for [FORMULA]. This was done by calculating the derivative of the logarithm of the likelihood function (S) with respect to [FORMULA], and equating this derivative to zero. The variation of the amplitudes, linewidth, central frequency and background noise using different values for [FORMULA] proved to be very small in the fitting and were neglected. The expression of [FORMULA] as a function of [FORMULA] is given by (see Appendix A):

[EQUATION]

where [FORMULA] and [FORMULA] are the nominal values of [FORMULA] and [FORMULA] respectively. Note that when [FORMULA], [FORMULA] will be equal to [FORMULA] and [FORMULA] is the maximum of the function. The continuous line in Fig. 4 is the non-linear squares fit of the theoretical calculation (Eq. 7) to the [FORMULA] weighted average (full circles), which gives: [FORMULA] nHz and [FORMULA]. Note that the estimated uncertainties in the fitted [FORMULA] coefficients for each [FORMULA] are correlated; however, as they are very similar, we used the same weight for all points in the fitting and the estimated [FORMULA] and [FORMULA] and their errors are an approximation. These values agree quite well with the Fourier fitting using the theoretical leakage matrix element [FORMULA]: [FORMULA] nHz.

As a conclusion, we can say that the quadratic dependence of the rotational splittings upon the leakage elements is another reason to fit the Fourier spectra instead of the power spectra. Although Eq. (7) is only an approximation, it could be used to calculate the splitting coefficient or at least its weighted average with more precision, independently of the correctness of the leakage within a multiplet. However, in the next section we are going to show that in our case [FORMULA] nHz is still biased because of modes aliasing with different degrees. We limited ourselves in this section to the analysis of [FORMULA] where the leakage between the modes is more critical.

3.3. Leakage between modes of different degrees

So far we have mentioned only the leakage between the elements of a given multiplet. However, it is also possible to have leakage between modes of different degrees, i.e. aliasing modes. The leakage will be due to modes with a degree similar to the target mode; modes with a very different degree will have a negligible amplitude in the target spectrum. On the other hand, we will be concerned by the leakage of modes that have a frequency close to that of the target mode in the range [FORMULA] 1500 to 4000 µHz. To identify the aliasing modes, we plotted the mode frequencies in an echelle diagram (Fig. 5). We can see that the [FORMULA] modes overlap in frequency with the [FORMULA] and [FORMULA] modes, [FORMULA] modes overlap with [FORMULA] modes and so on.

[FIGURE] Fig. 5. Echelle diagram centered on [FORMULA]. The even degree modes are indicated as full circles and the odd modes as empty circles. The size of the circles is proportional to its degree. The horizontal bar crossing each circle represents the splitting due to the solar rotation and has its approximate magnitude: [FORMULA] µHz. The horizontal bars on the top right are the intervals where the modes are fitted: 17 µHz for [FORMULA] and 22 µHz for [FORMULA]. The frequencies used here for [FORMULA] are the results of this paper (Sect. 4) and for [FORMULA] were obtained by the GONG project (Hill et al. 1996).

The influence of the leakage of [FORMULA] and 9 modes in the determination of the [FORMULA] splitting coefficients can be clearly seen in Fig. 3 as a step for modes with [FORMULA] µHz. However, as the mode degree increases, the fitting of the spectra becomes less sensitive to the leakage. In fact, for [FORMULA], it appears in the GONG data mainly as a second order effect: a bump in [FORMULA] coefficients (Fig. 6a). These higher degree modes have many more elements in the multiplet which enable a much better determination of their central frequencies and splitting coefficients, independently of the influence of the leakage, which is not, in the same way, present in the ([FORMULA]) observed spectra of the multiplet. Modes with a degree which is different from the target by more than 3 have small amplitudes and seem not to affect the fitting, except for [FORMULA]. The leakage of [FORMULA] into [FORMULA] and vice-versa, does not seem to affect the fitted parameters either. In addition, the leakage between [FORMULA] and [FORMULA] modes happens only at very high frequencies ([FORMULA] µHz) and is not taken into account.

[FIGURE] Fig. 6. a  Splitting coefficients for the GONG data. Only modes with [FORMULA] µHz were plotted. Note the step in the [FORMULA] coefficients for [FORMULA] at [FORMULA] mHz-1 and the bump in [FORMULA] coefficient for [FORMULA] modes at [FORMULA] mHz-1. b  Same as a , except that the splitting coefficients of modes inside the leakage frequency range were found using spectra cleaned from aliasing degrees. Both features in [FORMULA] and [FORMULA] disappeared.

A straightforward way to consider the l leakage is to use a more complete leakage matrix in the likelihood function. Unfortunately, this is an extremely computer intensive task, because we have to fit multiplets with different degrees at the same time. We have instead constructed a more complete leakage matrix and applied its inverse to the observed Fourier spectra as described in Paper I and II: [FORMULA] (Eq. 1), therefore obtaining the `cleaned ' spectra. This cleaning method produces spectra that are very nearly independent of each other, at least in making the leakage matrix close to the identity matrix. In this case, the two methods (power and Fourier fitting) will be identical (Paper I).

We constructed the leakage matrix for [FORMULA], 6 and 9 and used it to obtain their `cleaned' spectra. Fig. 7 presents the splitting coefficients found fitting these `cleaned' Fourier spectra (triangles), calculated in the same way as before (Fig. 3), except that here the leakage matrix is the identity matrix. The step mentioned earlier clearly disappeared.

[FIGURE] Fig. 7. Absolute difference between the [FORMULA] coefficients and its weighted mean over n for [FORMULA] fitting the observed Fourier spectra (circles) and the `cleaned' Fourier spectra (triangles). They are slightly displaced in frequency for a better visualization. The weighted average values fitting the observed and the `cleaned' Fourier spectra are, respectively: [FORMULA] nHz and [FORMULA] nHz (in sidereal units). The full circles are the same as in Fig. 3.

On the other hand, for [FORMULA] modes, the bump that appears in [FORMULA] (Fig. 6a) at a given lower turning point could be caused by a solar physical process that is insensitive to the direction of propagation of the oscillations, such as an anisotropic sound speed or magnetic field at this solar radius, which will make the even coefficients be nonzero in their presence (Hill et al. 1991). Unfortunately, this curious feature is due to the leakage of [FORMULA] into 4, 8 into 5, and 9 into 6, i.e. between modes with [FORMULA] and [FORMULA]. Although the leakage onto [FORMULA], 5 and 6 happens at different frequencies: 2456, 2914 and 3317 µHz respectively, they happen coincidently at the same lower turning point: [FORMULA]. Applying the inverse of the complete leakage matrix to each pair, as in [FORMULA], and fitting the `cleaned' spectra, the new splittings found no longer present the bump (Fig. 6b), proving that the bump is due to the leakage between the modes. Although there is still some structure around [FORMULA] mHz-1, it could be due to an imperfect knowledge of the theoretical leakage matrix.

Comparing the fitted parameters using `cleaned' and `uncleaned' spectra, we found that the [FORMULA] and [FORMULA] splitting coefficients are the same (their determinations are very poor for such low degree modes); [FORMULA], [FORMULA] and [FORMULA] splitting coefficients found fitting the `cleaned' spectra are better than the `uncleaned' ones inside the frequency interval of leakage in the sense that their values do not oscillates or else oscillate with smaller amplitudes (see Fig. 6b). The unperturbed frequencies are different mainly inside the leakage interval where their difference is less than [FORMULA] µHz (Fig. 8).

[FIGURE] Fig. 8. Differences of frequencies between those found fitting the `cleaned' spectra and those fitting the `uncleaned' spectra ([FORMULA]).

As was already shown, it is very important to take the leakage of modes of different degrees into account. Using the inverse of the leakage matrix for cleaning the data is a simple way to solve this problem. However, outside the leakage frequency range of a given mode it is better to use the originally observed spectra (`uncleaned') as it is less manipulated and there is no necessity to use the `cleaned' one. In Fig. 6b we plotted the parameters fitting the `uncleaned' spectra, except for modes with (cf. Fig. 6): [FORMULA] and [FORMULA] µHz; [FORMULA] and [FORMULA] µHz; [FORMULA] and [FORMULA] µHz; and [FORMULA] and [FORMULA] µHz where we used the `cleaned' spectra. Besides, the `cleaned' spectra have a higher background noise level than the `uncleaned', whereas the mode amplitudes remain basically the same. This is going to affect essentially the determination of [FORMULA] and 6 modes with high frequency ([FORMULA] µHz) where the signal-to-noise ratio is poor anyway.

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
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