- Assessing the model of the mode and noise covariance
- Assessing the statistical distribution of the parameters
- Assessing the precision of parameters.

(39) |

(40) |

The variance of the pixel noise is assumed to be the same for the pixels with the same shape. The mean of the pixel noise is 0. For the LOI with its 12 pixels, there are 3 different shapes giving 3 independent noises.

After generating the synthetic signals according to Eq. (39), the data are fitted by minimizing the likelihood of Eq. (32). Figure 1 shows an example of Fourier spectra generated synthetically. The typical signal-to-noise ratio in the power spectra is about 20-30. The frequency resolution is equivalent to 4 months of data. We performed 1000 simulations of the spectra.

Hereafter, we have investigated the
influence of a wrongly assumed leakage matrix on the fitted parameters
of *l* = 1. We made 100 realizations and change the leakage parameter
between *m*=-1 and *m*=+1 by % from a nominal value for the LOI of 0.45.
Figure 4 shows the influence of varying the assumed leakage
element on the fitted parameters. It is quite interesting to note
that the inferred central frequency is insensitive to mistakes in the
leakage matrix. The linewidth becomes underestimated when the error
is larger than 20%, while the amplitudes become overestimated. The most
important result is the fact that the systematic error made on the
splitting is not linear but quadratic. This systematic error can become as
large as the error bars. For example, with 1 year of LOI data and
averaging over 10 modes, the error bars on the mean splitting is about
15 nHz; this should be compared to a systematic error of 10 nHz for an
error of 10% of the *l*=1 leakage elements.

Another test similar to that of the *l*=1
was performed with the *l*=2 mode. We have assumed that
all the off-diagonal elements of the leakage matrix were wrong by the
same fixed amount. Figure
5 shows the results only for the splitting coefficients (from *a _{1}* to

It means that
it is quite easy to underestimate the splitting whenever we under- or
overestimate the leakage element. As a matter
of fact, this behaviour was also found in the GONG data for *l*=1 and
2 (Rabello-Soares & Appourchaux 1998, in preparation).
On the other hand, errors in the
leakage matrix will not result in overestimating the splitting. If
the splitting is overestimated, the most likely source should be the
presence of other degrees not taken into account in the analysis.

We also checked the
correlation of the splitting coefficients derived for *l* = 2.
Figures 6 and 7 show respectively the variance and
the covariance of the splitting coefficients as a function of the leakage elements error.
It can be concluded that the splitting coefficients become correlated
only when a large overestimation of about 50% is made for the
off-diagonal leakage elements. This result is only valid when
fitting Fourier spectra. For other methods, such as fitting power
spectra, possible correlation amongst the splitting coefficients could
have drastic consequences for the inverted solar rotation profiles.

Figure 4:
Influence of the fitted parameters to relative changes of the assumed
leakage element between m=-1 and m=+1 for l=1. (Top) Frequency,
splitting a, ; (Middle) (Amplitude) for
_{1}m=-1, 0, 1; (Bottom) (pixel noise). The target parameters are the
same as for Fig. 2. Please note the parabolic
shape for the splitting |

- Assuming that the 2
*l*+ 1 power spectra are independent of each other and are not influenced by*m*leaks, i.e. a single mode is present for a given*m*. This is the way the GONG data are commonly fitted. - Assuming that the 2
*l*+ 1 power spectra are independent of each other but are influenced by*m*leaks, i.e. we use only the diagonal of the mode covariance matrix. This is the way the LOI data were fitted by Appourchaux et al. (1995). - Assuming that the 2
*l*+ 1 Fourier spectra are dependent of each other and are influenced by*m*leaks. This is the way described in this paper after the work of Schou (1992).

Figure 8:
Comparison of low-degree splittings measured using 1000 Monte-Carlo
simulations by 3 different fitting techniques. The nominal splitting is
410 nHz. The + is the technique commonly used by GONG, the asterisk is the
technique used by Appourchaux et al. (1995), the diamond is the
technique described in this paper and also used for the SOI/MDI data. The
error bars are the formal error bars for a single realization |

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