## 2. Data analysis## 2.1. Maximum likelihood estimationWe used a maximum likelihood method for fitting the peaks in the observed Fourier spectra. We derived the amplitudes, linewidths, frequencies and noise characteristics represented by . Instead of fitting the power spectra () by minimizing the well-known expression given by Duvall & Harvey (1986), we have, in this work, fitted the Fourier spectra using the equation given by Paper I, and adapted from Schou (1992). The main difference between the two methods lies in the way they treat the spectrum correlation due to the mode leakage. The first method takes into account only the diagonal terms of the covariance matrix , assuming that the observed spectra are statistically independent from each other and have a probability density function with 2 degrees of freedom. However, from Eq. (1), the observed spectra are a linear combination of the single modes, i.e. they are statistically dependent upon one another and have a multivariate normal probability density function. In this method, the leakage between the modes is taken into account in a more complete way (for a detailed discussion see our previous work: Paper I and II). We have applied this fitting technique to 360 days of GONG data starting on 23 August 1995 with a 87 duty cycle. The GONG consists of a network of 6 sites with solar velocity imagers ( 250 250 pixels) located around the Earth to obtain nearly continuous observations of the Sun's five-minute oscillations (for a detailed description, see Harvey et al. 1996). After extracting a single () mode by spherical harmonic decomposition, an FFT of the whole time series obtained through the merging of all the sites data is computed. No sidebands are visible due to the high continuity in the data. Fig. 1 shows the power spectra of the time series for and modes.
A Lorentzian profile was used as a model of the variance of the spectrum of a single mode (), or, in a different notation, : where is the power amplitude, the full linewidth at half maximum and is the central frequency. These parameters plus the background noise () are the ones that we want to find. We used the same linewidth for the entire multiplet (Appourchaux et al. 1995; Rabello-Soares et al. 1997). Also, we used the same amplitude for the modes; it is expected that the prograde and retrogade modes have the same amplitude and, in fact, we made some tests where they proved to be very similar. The same background noise is fitted for the modes with . However, for higher degree modes, a simplification was made: the background noise was shown to be very similar for all s in a multiplet and the convergence of the fitting is easier using only one parameter for the multiplet. We kept the background noise parameters for low degree modes where the central frequency accuracy is more sensitive to the fitting. The frequency of the mode () is given by: where is the mean frequency of
the mode multiplet and are
orthogonal polynomials with degree We have fitted around the frequency of a given
() mode in a 17- ## 2.2. Leakage matrix computationAn important point in the p-mode parameter determination is the correct calculation of the leakage matrix (Eq. 1) in fitting the power or Fourier spectra. Following Schou & Brown (1994), the calculation of the leakage matrix of the GONG data was described in Paper II. The leakage matrix has some interesting properties. When
is odd, its elements are zero
because of the parity of the spherical harmonics. The leakage between
modes with different In Paper II, we developed and described a powerful test to
check the correctness of the calculated leakage elements by looking
directly in the observed Fourier spectra, namely the ## 2.3. Noise covariance matrix estimationThe noise covariance matrix
(Eq. 2) could be determined by an expression similar to the leakage
matrix (see Schou & Brown 1994 and Paper I). In fact its
elements, for which is odd, are zero
in the same way as for the leakage matrix elements (Papers I and II).
Another way to calculate the noise covariance matrix, and one which
will be used here, is to estimate it directly from the cross spectrum
outside the p-mode band. Unfortunately, the different source of solar
and instrumental noise introduces a frequency dependence in the noise
covariance elements. It is thus not obvious which part of the
frequency spectrum to use. As the p modes are equally spaced in
frequency, the noise covariance is calculated in between the modes
with radial order To calculate the noise covariance between the
Fourier spectra within a multiplet
with a given , we selected windows of
approximately 20 Instead of estimating the noise covariance directly, we computed the ratio of the noise covariance matrix because it varies slowly with frequency and we can, in most cases, fit a straight line to its frequency dependence: where the average is over the 20
Note that is the noise level of the observed power spectrum: . Thus, the noise covariance matrix is constructed using the estimated ratio cross spectrum: © European Southern Observatory (ESO) 1999 Online publication: April 28, 1999 |