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

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4. Comparison with IPHIR data

4.1. Extraction of 11 modes

Since the conclusions of Baudin et al. (1996) about 160 days of IPHIR data were obtained from time variations of the power instead of the energy, using a different normalization and a different time resolution, we have first re-analysed these data with the method described above, using the same 11 modes ([FORMULA], [FORMULA], and [FORMULA], [FORMULA]). The central frequency [FORMULA] is taken from Toutain & Fröhlich (1992). The higher level of noise limits the size of the filtering window to [FORMULA] Hz, leading to a time resolution of 1.9 days, and a statistical study on 82 points. We have removed the data surrounding two gaps in the series, around the 5th and the 61st day. The resulting sample is shortened to 78 points only.

According to Fig. 8, the distribution of energy of each of the 11 modes of IPHIR is compatible with an exponential distribution (apart from the mode [FORMULA], [FORMULA]).

[FIGURE] Fig. 8. Variance test (plus) and KS test (square) for the 11 modes [FORMULA], [FORMULA], and [FORMULA], [FORMULA] observed by IPHIR with a filtering window of 6 [FORMULA] Hz.

The correlation coefficient of the modes energy, two by two, is shown in Fig. 9. The mean value is [FORMULA], the statistical error being [FORMULA]. Altogether, [FORMULA] of the couples (2 out of 45 couples) present correlations contained, in absolute value, between 2 and 3 standard deviations, which is comparable to the [FORMULA] expected for a normal distribution of the statistical error.

[FIGURE] Fig. 9. Correlation coefficients of the 11 modes observed by IPHIR, two by two. For each value of n, the long and short ticks correspond to [FORMULA] and [FORMULA], respectively. The symbol [FORMULA] (resp. -) is used for a positive (resp. negative) correlation. The smallest symbols correspond to correlations smaller than the statistical error, intermediate and big symbols correspond to correlations smaller than 2 and 3 statistical errors respectively.

4.2. Test of the null hypothesis

We have also extracted the same modes, with the same filtering window, from the first 153 days of GOLF data to obtain a comparable sample of 78 points.

[FIGURE] Fig. 10a and b. Histogram (15 bins) of [FORMULA] and cumulative distribution, compared to a [FORMULA] distribution for GOLF (above) and IPHIR (below). In the IPHIR case, the dashed lines correpond to the theoretical distribution [FORMULA].

The difference between the two series appears on the distribution [FORMULA] shown in Fig. 10, where both the variance and the KS tests indicate that the modes are likely to be less independent in IPHIR data than in GOLF data.

The tests applied to GOLF data are compatible with the null hypothesis ([FORMULA], [FORMULA]), which is consistent with the results of Sect.  3.

By contrast, the same tests applied to IPHIR data reject the null hypothesis with a [FORMULA] confidence level with the variance test, and a [FORMULA] confidence level for the KS test ([FORMULA], [FORMULA]).

4.3. Test of the " [FORMULA] -hypothesis"

While the cumulative distribution [FORMULA] of GOLF does not show any systematical trend when compared to [FORMULA], the cumulative distribution [FORMULA] of IPHIR shows a clear trend. This trend is successfully suppressed when compared to the distribution [FORMULA] (Fig. 10). Fig. 11 shows that within our simple model, the signal of IPHIR would be absolutely normal as regards our tests ([FORMULA], [FORMULA]) if a fraction [FORMULA] of each mode energy were common to all the modes, correponding to a mean correlation [FORMULA].

[FIGURE] Fig. 11. Comparison of IPHIR data ([FORMULA] points) with a theoretical distribution made of [FORMULA] modes containing a fraction [FORMULA] of energy in common. The vertical dotted lines delimit the range of the correlation coefficient determined by Eq. (9). The KS test remain inside the upper [FORMULA] region for [FORMULA], corresponding to a fraction [FORMULA].

Error bars are obtained by varying the parameter [FORMULA]: the variance test leads to [FORMULA], while the KS test obtains [FORMULA].

Moreover, the correlation computed from Eq. (9) with the statistical error given by Eq. (B6) is [FORMULA]. Although this analytical estimate is less reliable than the tests based on Montecarlo simulations (Eq. (B6) neglects the error in estimating the mean energy of each mode), it is useful as a quick check of the results.

It is therefore comforting to notice, as can be seen in Fig. 11, that the range of correlations defined by these three methods overlap in the range [FORMULA], correponding to [FORMULA]. Of course, this overlapping region cannot be directly interpreted in terms of a standard deviation. We shall adopt the conservative range obtained with the KS test, which takes the full distribution into account: [FORMULA], corresponding to a fraction [FORMULA].

4.4. Additionnal checks

In order to check the possibility that the correlation might come from a multiplicative noise (such as due to a pointing noise), we have computed the correlation between 11 windows of noise centered [FORMULA] (resp. [FORMULA]) to the right of each mode. This test indicates that the noise itself is not correlated, with [FORMULA] and [FORMULA] (resp. [FORMULA] and [FORMULA]).

We have checked the effect of changing the size of the filtering window to 4 [FORMULA] Hz (no noise, but low statistics of 52 points) and 8 [FORMULA] Hz (good statistics of 106 points, but IPHIR is influenced by the noise). While a smaller filtering window still favours [FORMULA], a larger window takes into account a significant fraction of uncorrelated noise, as expected, resulting in a slightly lower value of [FORMULA].

One might also suspect that the discrepency between the IPHIR distribution [FORMULA] and a [FORMULA] distribution is due to the mode [FORMULA], [FORMULA] which is not well fitted by an exponential distribution (see Fig. 8). Nevertheless, performing the same analysis without this particular mode leads to the same conclusion: [FORMULA] and [FORMULA] if [FORMULA], while [FORMULA] and [FORMULA] if [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998