5 Conclusion

We have given a step by step recipe for fitting $(m,\nu)$ Fourier spectra. If one wants to implement similar fitting technique, one should compute, first the leakage matrices according to Eqs. (13) and (14), second the mode covariance matrices with Eq. (22) (or using Eq. (23)), third the noise covariance matrices with Eq. (27) using a model of solar noise, fourth compute the likelihood function using Eq. (32). The use of Monte-Carlo simulations will ensure the success of the implementation. Routines for fitting p-mode Fourier spectra are available as freeware on the VIRGO home page: virgo.so.estec.esa.nl; they are written in the IDL macro language.

Last but not least, Eq. (36) showed us the equivalence between fitting data for which the leakage matrix is not the identity, and fitting data for which it explicitly is. This last statement is true provided that we know perfectly well the leakage matrix. In this case the p-mode parameters fitted using MLE are not, or very weakly, biased, and have minimum variance. For instance, we showed for the splitting that other commonly fitting methods result either in a bias and/or larger formal errors. We have also studied the effect of an imperfect knowledge of the leakage matrix on the fitted parameters, in order to derive the effect of systematic errors on the most interesting parameters: the splitting and mode frequency. We found that the central frequency is insensitive to systematic errors in the leakage matrix, while the splitting coefficients (a_i) have a quadratic dependence upon those errors. These systematic errors will have influence on the inverted solar rotation profiles.

Finally, we would like to stress again that the correct statistical treatment of the p-mode data is of vital importance for deducing unbiased p-mode parameters.

Acknowledgements

Many thanks to Takashi Sekii for constructive comments on the manuscript, and for extensive cyberspace chatting :-). We are grateful to the referees for their constructive comments.