## Appendix A: extraction of the time evolution of the energyLet 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 modesThe 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. 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 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 Let us now show that the probability density
of also varies like
, to first order. The moment of order The fact that every moment varies at least like (Eq. B11) implies: The only function satisfying Eq. (B15)
for any value of 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 |