## 3. Source of systematic errors## 3.1. Fourier versus power spectra fittingIn 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 are slightly overestimated with a bias of about 10 nHz. Besides, the uncertainties of the fitted splitting coefficients for modes are larger when fitting the power spectra and similar for (see Fig. 8 in Paper I). These predict quite well the differences between the two techniques when applied to the observations presented here. For , the splitting coefficients
found fitting the power spectra present a larger range of values and
their weighted average over
We can conclude that the Fourier fitting is very important in determining mode parameters and still important for 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 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 determinationThe 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 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 , there will be only one matrix element to be determined, i.e. the leakage between and , which we called . We undertook the same exercise here, using values of different from that of the theoretical value (Sect. 2.2), the 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 coefficients calculated using the power spectra vary almost linearly with , those found using the Fourier spectra obey a quadratic law. Thus, when fitting the Fourier spectra, the coefficient could only be underestimated using an incorrect value for ; its correct value corresponds to the maximum value of the coefficient. Although these variations in the splitting coefficient are small in comparison to their error bars (around 15 nHz for an error of 20 in ), they are significant because of their systematic nature.
We have analytically derived the dependence of
upon
fitting the Fourier spectra for
. This was done by calculating the
derivative of the logarithm of the likelihood function ( where and are the nominal values of and respectively. Note that when , will be equal to and 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 weighted average (full circles), which gives: nHz and . Note that the estimated uncertainties in the fitted coefficients for each are correlated; however, as they are very similar, we used the same weight for all points in the fitting and the estimated and and their errors are an approximation. These values agree quite well with the Fourier fitting using the theoretical leakage matrix element : 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 nHz is still biased because of modes aliasing with different degrees. We limited ourselves in this section to the analysis of where the leakage between the modes is more critical. ## 3.3. Leakage between modes of different degreesSo 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 1500 to 4000
The influence of the leakage of
and 9 modes in the determination of the
splitting coefficients can be clearly
seen in Fig. 3 as a step for modes with
A straightforward way to consider the We constructed the leakage matrix for , 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.
On the other hand, for modes, the
bump that appears in (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 into 4, 8
into 5, and 9 into 6, i.e. between modes with
and
. Although the leakage onto
, 5 and 6 happens at different
frequencies: 2456, 2914 and 3317 Comparing the fitted parameters using `cleaned' and `uncleaned'
spectra, we found that the and
splitting coefficients are the same
(their determinations are very poor for such low degree modes);
,
and 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
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):
and
© European Southern Observatory (ESO) 1999 Online publication: April 28, 1999 |