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

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Appendix A: extraction of the time evolution of the energy

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

From Eq. (1), the non-zero Fourier components of [FORMULA] (and [FORMULA]) correspond to frequencies between [FORMULA] and [FORMULA]. If, as is usually the case, [FORMULA], the high frequency oscillations of [FORMULA] are well separated from the slower variations of [FORMULA] and [FORMULA], 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 [FORMULA], we assume that the velocity residual of each mode [FORMULA] is made of a superposition of two independent signals, where [FORMULA] is common to all the modes, and all the [FORMULA], [FORMULA], are independent.
Using the filtering method described in Sect. 1, we introduce the parameter [FORMULA] as:

[EQUATION]

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

[EQUATION]

The correlation between two modes [FORMULA] is then:

[EQUATION]

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

The sum [FORMULA] of the normalized energies is defined as:

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

The moment of order l is defined as:

[EQUATION]

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

[EQUATION]

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

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

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© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998
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