The energy integrated over a time interval, i.e. the power of the mode, was computed by Chaplin et al. (1995) using a Fourier transform over short subseries. More sophisticated methods based on the wavelet analysis were developped by Baudin, Gabriel & Gibert (1994) in order to analyse the variations of power both with time and frequency.
Frequency resolution is not required for our study. Since the distribution of energy is likely to be mathematically simpler than the distribution of power (Kumar, Franklin & Goldreich 1988), we have prefered to extract the energy directly.
Let be the oscillatory velocity (e.g. integrated over the surface of the sun), filtered in the Fourier domain through two windows of width centred on the eigenfrequencies . Its Fourier transform is therefore equal to zero out of these windows. The time evolution of the energy of this isolated mode can be obtained by a bivariate spectral analysis, as in Toutain & Fröhlich (1992). Here we favour a simpler method based on the inverse Fourier transform of the line, translated around . It is shown in Appendix A that the energy of this mode can be written as follows:
This approach is equivalent to the one used by Chang & Gough
(1995), by means of the Hilbert transform of
the velocity, since .
Denoting by T the total length of the observation, the frequency resolution of the Fourier transform is , and the filtering window contains points. The inverse FFT algorithm is used to compute Eq. (1) and define the energy at p successive instants. Eq. (3) then guarantees that the resulting energy is not oversampled.
We have considered the set of p modes corresponding to , and 1, between 11th April 1996 and 14th February 1997 (a publication concerning the calibration procedure is in preparation). The Fourier transform of the resulting velocity over these 310 days allows a filtering window size of ( days) for this set of modes. The window is symmetric with respect to the centroid of the line, , which is determined according to Lazrek et al. (1997). The two m -components of the mode , however, are not separated. In contrast with IPHIR, the width of the window is determined by the proximity of another mode (), rather than by the level of noise which is here very low.
Fig. 1 shows the time evolution of the energy of the 18 selected modes and , normalized to their mean energy. The GOLF instrument was stopped for one day on 8th September 1996. Four days of signal were removed from our statistical study (around the 156th day on Fig. 1) in order to account for the stabilisation of the instrument. The resulting sample is made up of 210 points.
Following the picture of a thermodynamic equilibrium between the random motions of the convective cells and the oscillating cavity (Goldreich & Keeley 1977), we wish to compare the observed sample of energies , with an exponential distribution. Any exponential distribution is defined by a single parameter, its mean value m. Fig. 2 shows a typical histogram and cumulative distribution for the modes extracted from the GOLF data (the cumulative distribution is defined as the primitive of the density of probability, it increases monotonially from 0 to 1). They are compared to an exponential distribution whose mean value is estimated from the sample of p points. Using the Maximum Likelihood approach, the best unbiased estimator of m for an exponential distribution is the following:
(i) The variance test
The variance of an exponential distribution coincides with the square of its mean value. We check this property by computing, for each mode of the GOLF data, the ratio of the estimated variance (denoted by ) to the estimated mean value squared :
Each value is interpreted owing to the cumulative distribution of , obtained if were built from a true exponential distribution. is computed numerically using a Montecarlo method, with exponential samples of p points. For each of the modes selected, is the fraction of these trials leading to a value of larger than the one observed. Since we are interested only in knowing whether the observed is typical of an exponential distribution or not, we shall give equal importance to the lowest and highest values of the variance by measuring the quantity .
(ii) The Kolmogorov-Smirnov test
Instead of doing this, we have used a Montecarlo method of samples in order to define the cumulative distribution of the distance . therefore indicates the fraction of these trials leading to a distance larger than the value observed.
For each of the modes selected, a value of close to would indicate that the observed distribution is too far from the theoretical one. A value of close to is just as improbable, but would indicate an exceptionnal agreement between the theoretical distribution and the observed one.
(iii) Autocorrelation of the artificial exponential distributions
used in the Montecarlo method.
The output of the tests, however, is only slightly modified if distributions made of independent points are used.
For both tests, Fig. 3 shows a very good agreement for the set of modes selected. As an exception, the energy of the mode , is not exponentially distributed (, ).
Although the global shape of this distribution can be made compatible with an exponential distribution by adopting a mean value smaller than the estimated value ( for ), its variance is too large to be reconciled with the variance of an exponential distribution.
We have also analysed the distribution build with the 18 modes altogether (each mode is normalized by its mean energy). Even with this improved statistics of points, the variance and KS tests have not detected any significant deviation from an exponential distribution (, ).
© European Southern Observatory (ESO) 1998
Online publication: January 8, 1998