Astron. Astrophys. 355, 743-750 (2000)
6. Bias introduced when fitting a Lorentzian profile to an asymmetric resonance
6.1. Determination of a bias uncertainty
The uncertainty on a bias (frequency, line width or splitting)
cannot be calculated from those corresponding to the quantities
delivered by the two kinds of fits, because these uncertainties are
not dependent, the stochastic excitation being the same in the two
cases. To compute a bias uncertainty, we perform Monte Carlo
simulations: for each radial order and mode pair, we simulate 200
stochastic realisations of a Nigam's profile. For each realisation, we
do a Nigam and a Lorentzian fit. Then, for each mode parameter, the
histogram of the difference between the two results gives the
uncertainty we require. Note that the uncertainty on a bias can be
significantly smaller than the uncertainty on the absolute value of
the mode parameters (see Figs. 7 to 12).
Fig. 7. The frequency difference between asymmetric and Lorentzian fittings as a function of frequency and mode degree. The errors bars are computed by Monte Carlo simulations as described in the text.
Fig. 8. The linewidth difference between asymmetric and Lorentzian fittings as a function of frequency and mode degree.
Fig. 9. Determination of the linewidth using Lorentzian profiles.
Fig. 10. Determination of the linewidth using the Nigam formula.
Fig. 11. The splitting difference between asymmetric fitting and Lorentzian fitting, as a function of frequency and mode degree.
Fig. 12. Determination of the splitting (synodic) using the Nigam formula. Results for higher frequencies have larger errors bars and are not shown here.
6.2. Bias on the frequency
The frequency difference between symmetric and asymmetric fits is
plotted versus the frequency in Fig. 7 for the different modes.
There is a systematic shift for all the modes: the frequency infered
from fitting a Lorentzian profile is lower than those using an
asymmetric profile. The mean value of the bias is about 42 nHz for
=0, 85 nHz for
=1, 54 nHz for
=2 and 24 nHz for
=3. Not only the bias but also its
variation with frequency is an important input for the inversion
because the difference between frequencies has more impact on the
deduced solar structure than frequencies themselves. The difference
between the behaviour of =1 and the
other degree modes can be understood and reproduced in simulations
(Lorentzian fitting on pure Nigam profiles).
=1 behaves as an isolated mode (see
Sect. 5.1) and is influenced not only by B but
predominantly by . For
1, the influence of the neighbouring
mode dominates. The frequencies inferred from asymmetric fitting are
given in Table 1.
Table 1. Frequency and errors in µHz derived from asymmetric fitting for modes =0, =1, =2 and =3 with radial orders from n=12 to n=25. The observation duration is T=805 days.
6.3. Bias on the linewidth
The width difference between symmetric and asymmetric fitting
depends on the mode degree and on the frequency (see Fig. 8): the
widths inferred from a Lorentzian profile are lower than those from an
asymmetric profile for degree =1 and
=0 and inversely for
=3 the sign is frequency dependent.
The same simulations than those mentioned in the previous subsection
show that the influence of the components on each other in a multiplet
is dominant for the sign of the bias. The evolution with the radial
order is, as for the frequencies, influenced by both the line width
and the neighbouring mode.
The importance of this is such that when the asymmetry is taken
into account, the dispersion of the width with the degree is
considerably reduced as shown by Figs. 9 and 10.
6.4. Bias on the splitting
The difference between determinations of the rotational splitting
based on Lorentzian or asymmetric profiles depends on the degree (see
Fig. 11) and the difference increases with increasing frequency
for modes =1 and
=2. As for the width, the sign of the
bias depends on the degree, which we also confirmed by the
simulations. The Fig. 12 shows the results obtained and
Table 2 gives mean values over frequency for the two kinds of
fits. Note from Table 2 that, as we would expect, the uncertainty
induced by the stochastic excitation is not reduced when using an
asymmetric fit, but the dispersion with the degree is somewhat reduced
by a more valid fit. In particular, the relatively low value for the
splitting of the =2, observed here in
the Lorentzian case and reported in some publications (Lazrek et al.
1997, Gizon et al. 1996), is no longer obtained in the asymmetric
case. Moreover, the use of the Nigam formula permits a good
determination of the splitting farther in the high frequency range
than with a Lorentzian formula.
Table 2. Average and standard error in nHz of the synodic splitting over n when using asymmetric or Lorentzian fitting.
© European Southern Observatory (ESO) 2000
Online publication: March 9, 2000