4. Statistical behaviour
The spectral characteristics of the time series of 0.5-year coordinates in the source-fixed frames are investigated using the Allan variance method (Allan 1966, see a review of these methods in Rutman 1978). The Allan variance analysis allows one to characterize the power spectrum of the variabilities in time series, for sampling times ranging from the initial interval of the series to 1/4 to 1/3 of the data span, in our case 6 months to 3-4 years. This method identifies white noise (spectral density S independent of frequency f), flicker noise (S proportional to f-1), and random walk (S proportional to f-2), and it allows to specify the time frame in which a given type of noise is valid. Note that one can simulate flicker noise in a time series by introducing steps with a random amplitude at random dates. Fig. 3 gives the Allan variance graphs in the sources-fixed frames. In this log-log plot of the Allan variance as a function of sampling times (here 0.5, 1, 2, and 4 years), a slope equal to -1 is the signature of white noise, while a slope equal to zero is the signature of flicker noise. The blue/heavy stars plots are for the direction of the local x-axis, yellow/light open circles for the y-axis.
Most sources show similar stability sigantures in both the x- and y-directions up to four years sampling time. Four sources, 1611+343, 1633+382, 1739+522, and 1803+784, show a remarkable discrepancy between flicker noise behaviour in the x-direction and marked white noise behaviour, at a much lower level, in the y-direction.
Considering the Standard Model (deceleration parameter =0.5, cosmological constant =0) and a value =60km/s/Mpc for the Hubble constant, one can compute distances of objects as a function of the redshift (Mattig 1958), and hence transform angular dimensions into linear ones (Theureau 2000). Table 4 summarizes the characterization of the sources stability in angular and linear units.
Table 4. Source stability along maximum variance axis
© European Southern Observatory (ESO) 2000
Online publication: July 13, 2000