Astron. Astrophys. 333, 362-368 (1998)

3. Frequencies error bars results

The theoretical frequencies error-bars are based on Eq. (6) (Toutain and Appourchaux, 1994):

being the linewidth of the mode, T the duration of the observation, d the duty cycle and a function of the noise to signal ratio. This formula only applies to a singlet () mode. For multiplet modes (), if the known splitting components visible by the full disk observations were completely uncorrelated, they would, in principle, allow independent determinations of the same central frequency and subsequently decrease the error-bar of Eq. (6) like:

We took this quantity as reference for the comparisons.

3.1. IRIS data, sidelobes

Figs. 4 and 5 show a comparison between the former theoretical error-bars and the computed ones, computed on the 1990 dataset of the IRIS data with a bin in the spectral density of Hz. In all the cases, all the errors range from Hz to Hz. All the curves show a frequency dependence, related to the presence of the linewidth in the numerator of Eq.  6. Nevertheless, the error-bar curves are not identical to the width curves because they are also weighted by the SNR of the peaks which behaves differently.

Fig. 4a and b. and frequencies error bars for the year 1990, showing a good agreement between the Monte Carlo results and theory.

Fig. 5a and b. and frequencies error bars in year 1990. The MC results give higher numbers than the theory, showing the difficulty to observe the triplet and the quadruplet.

The presence of sidelobes of one day in the spectral density biases the measurement of the linewidths above 3.7 mHz, affecting both the computed and theoretical curves in the same way, so that they continue to match, up to 4.2 mHz. The presence of the daily sidelobes, taken into account by the fit, also affects the quality of the result in degrading by a small amount the frequency of a given peak, but in degrading its immediate neighbours at about 11.57 µHz, split or unsplit.

and degrade themselves reciprocally, although is more affected, being of poorer SNR, and degrades . Subsequently, we expect as a general trend to have smaller error bars on and than on the two others, and this is actually the case.

As to the hypothesis of the gain in precision afforded by the multiplets, there is a difference between Fig. 4 and 5. For , the relation seems verified and the MC numbers are closer to the of Eq. 7than to . It is not the case for or 3 where the MC values are closer to . We think that the precision in the central frequency of a multiplet can benefit from independent determinations on each component only when those components are unambiguously identified by the fit for most of our 400 statistical realisations. The situation is then easier for because of its high relative amplitude and the cleanliness of its surroundings, than for , affected by the left sidelobe of . Also we shall see that the split components do interfere each other in the complex space, leading to interferences in the spectral density that can wipe out completely one or several components of the multiplet. The higher the number of components, the higher the probability of interferences leading to a misidentification of the individual components by the fit.

3.2. GONG data, no sidelobes

From Figs. 4 and 5, it is not clear which effect plays the major role, although we suspect that the sidelobe pollution has a stronger effect than the 'blurring' of the splitting.

For this reason, we also performed MC simulations using 1 month of GONG data with a 98% duty cycle and a better SNR than the IRIS data (but a lower resolution). Fig. 6a-c shows the difference between MC and theoretical uncertainties for singlet, doublet and triplet modes. In this case, free from sidelobes, we can really size the effect of the splitting on the error bars: we move from an almost perfect match for to a significant discrepancy for the doublet and triplet modes. We conclude from this that no gain in precision is to be expected from the split nature of a peak and that error-bars computed from an formula are a good guideline for all the modes. Fig. 7 shows one example of two possibles realizations of a doublet generated with the same set of initial parameters, and one can see that a maximum likelihood fit will give two incompatible values of splitting, and that the frequency determination will be greatly affected.

Fig. 6a-c. Frequency error bars on GONG 1-month data for a singlet (a), a doublet (b) and a triplet (c). The frequency bin of the data is 0.34 µHz.

Fig. 7a and b. Splitting 'blurring'. Two Monte Carlo simulations of (l=1, n=16) for the year 1990 showing two differents realizations of the doublet.

© European Southern Observatory (ESO) 1998

Online publication: April 15, 1998
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