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

## Appendix A: extraction of the time evolution of the energy

Let us consider the real displacement filtered through a double window with a width , centred on the frequencies :

where is the complex conjugate of the first term. By analogy with an oscillator of eigenfrequency , the energy is defined as the sum of a kinetic and a potential part:

From Eq. (1) and (A1), the filtered displacement and velocity can be written as follows:

We first deduce from the relation that and are related as follows:

From Eq. (1), the non-zero Fourier components of (and ) correspond to frequencies between and . If, as is usually the case, , the high frequency oscillations of are well separated from the slower variations of and , and Eq. (A5) is approximated by Eq. (2).

## Appendix B: one parameter model of correlated modes

The amount of energy which is coherent among the modes can be estimated by constructing a simple one-parameter model as follows. Indexing the modes by , we assume that the velocity residual of each mode is made of a superposition of two independent signals, where is common to all the modes, and all the , , are independent.
Using the filtering method described in Sect. 1, we introduce the parameter as:

where , , are independent normalized normal distributions. The quantity can be interpreted as the ratio of the energy in the common signal to the total energy of the mode. We denote by the energy of the signal filtered in the Fourier space, normalized to its mean value:

The correlation between two modes is then:

For the sake of simplicity, is assumed to be independent of the mode j. This is equivalent to assuming that the correlation is uniform among the modes.

The sum of the normalized energies is defined as:

The variance of the estimator of the variance depends on the fourth moment of the distribution, and is equal to:

Let us show that the higher order moments of the distribution vary like , to first order.

where denotes the expectation value of the distribution. As the transformation does not change the distribution defined by Eq. (B3), only the even powers of can contribute to . It can therefore be expanded into powers of . Let us develop the product in Eq. (B7) and prove that the term of order is zero. We use below the special relation between the centred moments of a -distribution of order k:

Let us now show that the probability density of also varies like , to first order.
Given the Eqs. (B3)-(B5) defining , we can expand in powers of .

The moment of order l is defined as:

The fact that every moment varies at least like (Eq.  B11) implies:

The only function satisfying Eq. (B15) for any value of l is . Therefore both and its primitive (i.e. the cumulative distribution of involved in the KS test) vary like to first order.

We conclude that the sensitivity of the KS test is comparable to the sensitivity of the variance test.

© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998