LONG TERM STABILITY PREDICTION OF MILLISECOND 
                  PULSAR TIMINGS BY MULTI-VARIANCE ANALYSIS

                    F. Vernotte, G. Zalamansky, M. McHugh
                          Observatoire de Besancon
    41 bis, avenue de l'Observatoire - BP 1615 - 25010 BESANCON Cedex
Phone : 81.66.69.22 - Fax : 81.66.69.44 - E-mail : FRANCOIS@OBS-BESANCON.FR

The measurement of the time stability of the millisecond pulsars is of
importance for metrology as well as cosmology. Since pulsar timings are 
irregularly spaced, the usual tools of time-frequence metrology (variances) 
cannot be directly used in this case. However, these methods may be adapted.
 
In the case of regularly spaced data, the concept of structure functions [1],
which is an extension of the variance approach, is useful to determine the
variance (the structure function) which is optimized for a type of noise
and for an order of drift : e. g. a variance corresponding to a second order
derivative operator (as the Picinbono variance) can measure the level of a
low frequency noise but is non-sensitive to a linear drift. However, it is
not possible to define a variance simultaneously optimized for all types of
noise and for all order of drift.

The multi-variance method was developed to use different variances over the
same signal. It is then possible to select a set of variances in which each
variance is optimized to the determination of one parameter (of one noise
level, drift, or cut-off frequency). An appropriate choice of variances
yields a better estimation of the noise levels and may solve the ambiguity
between real drifts and very long term fluctuations.

Recently, we adapted this method to irregularly spaced timing data. In this
connection, we replaced the structure functions by an other method of
spectral density estimation : the lowest mode estimator, introduced by J.E.
Deeter and P. E. Boynton for the analysis of pulsar timing data [2]. This
method is based upon the construction of a set (as much as the number of
data) of orthonormal polynomials. This set depends on the time distribution
of the data. Since a characteristic frequency is associated to each
polynomial (as the maximum of its Fourier transform), each polynomial may be
considered as an estimator of the spectral density for its characteristic
frequency. Moreover, different set of polynomials can be constructed
according to two priorities : the order of drifts that must be removed and
the type of noise for which the sensitivity must be maximum. Thus, a
multi-variance system is developed using different sets of polynomials. The
details of this method are described and the results for different signals
are discussed in this paper.
[1]     W. C. Lindsey and C. H. Chie, 1976, Proc. IEEE, vol. 64, pp. 1652
[2]     J. E. Deeter and P. E. Boynton, The Astrophysical Journal, 1982,
        vol. 261, pp. 337