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Astron. Astrophys. 330, 341-350 (1998)

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2. Time evolution of the energy of a single mode

2.1. Method of extraction of the energy

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 [FORMULA] be the oscillatory velocity (e.g. integrated over the surface of the sun), filtered in the Fourier domain through two windows of width [FORMULA] centred on the eigenfrequencies [FORMULA]. Its Fourier transform [FORMULA] 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 [FORMULA] of the line, translated around [FORMULA]. 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 [FORMULA] of the velocity, since [FORMULA].
If the distribution of velocities is gaussian, then the real and imaginary parts of the function [FORMULA] are two independent gaussian distributions with identical amplitudes and variances. Thus Eq. (2) directly implies that the distribution of the energy is exponential, as expected.

Eq. (1) shows clearly that the time resolution [FORMULA] of the energy, reconstructed by Eq. (2), is related to the size [FORMULA] of the filtering window:


Denoting by T the total length of the observation, the frequency resolution of the Fourier transform is [FORMULA], and the filtering window [FORMULA] contains [FORMULA] 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.

2.2. Application to the GOLF data

We have considered the set of p modes corresponding to [FORMULA], [FORMULA] 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 [FORMULA] ([FORMULA] days) for this set of modes. The window is symmetric with respect to the centroid of the line, [FORMULA], which is determined according to Lazrek et al. (1997). The two m -components of the mode [FORMULA], however, are not separated. In contrast with IPHIR, the width of the window is determined by the proximity of another mode ([FORMULA]), 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 [FORMULA] and [FORMULA], 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.

[FIGURE] Fig. 1. Time evolution of the energy of the modes [FORMULA], [FORMULA] (above) and [FORMULA] (below). The energy of each mode is normalized to its mean value.

2.3. Statistical tests

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 [FORMULA], 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 [FORMULA] 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:

[FIGURE] Fig. 2. Histogram (20 bins) of the energy of the mode [FORMULA], [FORMULA], and its cumulative distribution, compared to a theoretical exponential distribution. The variance test [FORMULA] compares the observed variance to the theoretical one, while the Kolmogorov-Smirnov test [FORMULA] depends on the maximum distance between the theoretical and the observed cumulative distributions.


(i) The variance test
A simple test consists in checking that the first moments of the distribution (mean value and variance) are compatible with those of a theoretical exponential distribution.

The variance [FORMULA] 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 [FORMULA] of the estimated variance (denoted by [FORMULA]) to the estimated mean value squared [FORMULA]:


Each value is interpreted owing to the cumulative distribution [FORMULA] of [FORMULA], obtained if [FORMULA] were built from a true exponential distribution. [FORMULA] is computed numerically using a Montecarlo method, with [FORMULA] exponential samples of p points. For each of the modes selected, [FORMULA] is the fraction of these [FORMULA] trials leading to a value of [FORMULA] larger than the one observed. Since we are interested only in knowing whether the observed [FORMULA] 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 [FORMULA].

(ii) The Kolmogorov-Smirnov test
While the variance test depends only on one particular moment of the observed distribution [FORMULA], a more global comparison is achieved with the Kolmogorov-Smirnov (KS) test on the cumulative distribution [FORMULA]. This test measures the maximum distance [FORMULA] between [FORMULA] and a theoretical exponential cumulative distribution [FORMULA]. If the mode energies were exponentially distributed, the statistics of [FORMULA] would be described by a cumulative distribution denoted by [FORMULA]. Since the mean value m of the reference ditribution is estimated from the data, we cannot use the standard formulae (Numerical Recipies 1992, Chapt. 14.3) to fit [FORMULA].

Instead of doing this, we have used a Montecarlo method of [FORMULA] samples in order to define the cumulative distribution [FORMULA] of the distance [FORMULA]. [FORMULA] therefore indicates the fraction of these [FORMULA] trials leading to a distance [FORMULA] larger than the value observed.

For each of the modes selected, a value of [FORMULA] close to [FORMULA] would indicate that the observed distribution is too far from the theoretical one. A value of [FORMULA] close to [FORMULA] 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.
All of the p modes selected are autocorrelated over a timescale comparable to their damping time (2 to 4 days), usually deduced from the Full Width at Half Maximum (FWHM) of their lorentzian fit in the Fourier space. For the sake of accuracy, we have therefore used exponential distributions with comparable autocorrelation in order to compute the theoretical cumulative distributions [FORMULA] and [FORMULA] in our Montecarlo simulations. Each one is obtained by first creating a time series of a damped oscillator excited by a Gaussian noise, and then extracting the energy with the method described in Sect.  2.1. The damping time of the oscillator is chosen such that it corresponds to a FWHM of [FORMULA] in the Fourier space.

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 [FORMULA], [FORMULA] is not exponentially distributed ([FORMULA], [FORMULA]).

[FIGURE] Fig. 3. Result of the Kolmogorov-Smirnov test (square) and variance test (plus) of the modes [FORMULA] and [FORMULA], [FORMULA]. As a reference, the horizontal dashed lines delimit the upper regions which should contain [FORMULA], [FORMULA], [FORMULA] of the events, respectively.

Although the global shape of this distribution can be made compatible with an exponential distribution by adopting a mean value [FORMULA] smaller than the estimated value ([FORMULA] for [FORMULA]), 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 [FORMULA] points, the variance and KS tests have not detected any significant deviation from an exponential distribution ([FORMULA], [FORMULA]).

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© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998