(9) |

(10) |

(11) |

In summary, to understand the statistics of resolved observation, one has to follow four steps:

- Compute leakage matrices,
- Compute mode covariance matrices (related to the leakage),
- Compute noise covariance matrices,
- Generate the likelihood from the theoretical probability distribution.

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

(18) |

If the
weight functions *W*_{l,m} and the observed perturbations have the
same symmetry properties as the spherical harmonics *Y*_{l,m} (or if
), the leakage
matrix is real as shown by Schou (1992). In addition the
leakage elements of are zero if *l*+*m*+*l*'+*m*'
is odd; this is the case when the Sun is *not* tilted with respect
to the observer's axis of reference (*P* = 0, *B* = 0). If the axes of reference
of *W*_{l,m} differ
from that of the *Y*_{l,m} these 2 properties can be lost. For
instance, an incorrect orientation of the Sun axis with respect to
the detector axis could lead to a complex leakage matrix; or a Sun
seen at an angle give a real leakage matrix with non-zero elements with *l*+*m*+*l*'+*m*'
odd. This latter property has been used by Gizon et al. (1997)
to infer the inclination of the Sun's core.

Equation (13) is valid when the size of the pixel is small compared with the
spatial scale of the degree. When the pixels are larger, one should
write the following:

(19) |

(20) |

(21) |

(22) |

Schou (1992)
gave an equation similar to Eq. (22) for a real leakage matrix
and for a single degree. Here we add a subtlety
to the formulation of Schou (1992), the matrix
can be decomposed as follows:

(23) |

(24) |

(25) |

(26) |

(27) |

(28) |

(29) |

Again, when the size of the pixel is large compared with the
spatial scale of the degree, Eq. (27) is rewritten as
follows:

(30) |

(31) |

Using Eq. (31), we can write the likelihood *L* of an observation
of at *N* different frequencies
as given by:

(32) |

In principle, given the observed vector , it is always possible in
the absence of noise to recover the vector . Due to the presence
of noise only a solution close to the ideal one can be found that
will minimize the correlation between the components. Provided that
the leakage matrix can be inverted, we have by analogy to Eq.
(12):

(33) |

(34) |

(35) |

(36) |

(37) |

(38) |

It can be derived from Eq. (37) that it is also possible to remove correlation due to the noise by replacing by a proper matrix associated with . The derivation of this matrix is given in Appendix A.

For the other low degree modes, the likelihood
becomes somewhat more complicated. It is well known, that in the diagramme of *l*=1, there are
leaks coming from other degrees. The *l*=6 and *l*=9 modes overlap with the *l*=1, while the *l*=3 modes overlap only at higher frequencies when the linewidth is larger than about 5 Hz. In the diagramme of *l*=4, there are
leaks of *l*=7 and vice versa (Appourchaux et al. 1997). The
leaks have severe effects on determination of the p-mode parameters of the
*l*=1. When many degrees are overlapping, one should use Eq. (32)
using the covariance matrix for *l* and *l*'. Nevertheless, we do not
advice to do so for fitting the p modes; it has some severe computer speed
penalty. Instead we advice to clean the data by inverting the full leakage
matrix taking into account the effects of the various degrees on each
other, in a similar way to Eq. (33). For example for *l*=1, one
should compute the leakage matrix on a sub-space of degrees namely for *l*=1, 6 and 9. These 3 degrees interact strongly in the Fourier spectra. For *l*=4 and *l*=5 one should compute the leakage matrix on sub-spaces for *l*=4 and 7, and for *l*=5 and 8. The advantage of this direct cleaning is to replace the original aliasing degrees by new aliasing degrees which are further away, in frequency, from the modes to be fitted. This technique has been applied to the LOI and GONG data, and is developed in Part II.

Last but not least, when the signal-to-noise ratio is high (i.e. we neglect in Eq. (37)), the elements of the vector are all independent of each other, leading to a statistical distribution which is a product of with 2 degree of freedom. This is an approximation which is useful and less incorrect that using this statistics for the GONG data for the vector as in Hill et al. (1996).

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