The art of fitting p-mode spectra - I. Maximum likelihood estimation

In this article we present our state of the art of tting helioseismic p-mode spectra. We give a step by step recipe for tting the spectra: statistics of the spectra both for spatially unresolved and resolved data, the use of Maximum Likelihood estimates, the statistics of the p- mode parameters, the use of Monte-Carlo simulation and the signicance of tted parameters. The recipe is applied to synthetic low-resolution data, similar to those of the LOI, using Monte-Carlo simulations. For such spatially resolved data, the statistics of the Fourier spectrum is as- sumed to be a multi-normal distribution; the statistics of the power spectrum is not a 2 with 2 degrees of free- dom. Results for l = 1 shows that all parameters describ- ing the p modes can be obtained with negligible bias and with minimum variance provided that the leakage matrix is known. Systematic errors due to an imperfect knowl- edge of the leakage matrix are derived for all the p-mode parameters.