Statistical analysis of the time series of permanent GPS stations

SUMMARY Time-series of the horizontal coordinates of 21 GPS stations of the EUREF Permanent Network in the Alpine Mediterranean area with three or more years of continuous tracking have been computed with the intent of estimating velocities and their uncertainties, taking into account the detailed structure of their noise. The power spectral densities demonstrate that coloured noise, mostly flicker (1/f ) noise, can be present at frequencies below 6 cycle yr −1 , while at higher frequencies the spectrum tends to a regime of white (i.e. frequency-independent) noise. This statistical information is used to obtain more accurate estimates of station velocities and of their uncertainties than by the standard least-squares method. Following an approach well known in the analysis of time-series of frequency standards, the stability of each time-series is computed as a function of time, in the sense of a two-sample Allan variance. The power spectral density of the time-series is used to infer the variance of the change in the slope, with 1σ probability, of two consecutive, equal-length batches of a given time-series, and this as a function of the length of the batch. The power spectral density of each time-series is then converted into an autocorrelation function. Taking into account the correct correlations of pairs of samples as a function of their lag, the slope of each time-series is estimated by the least-squares method, with a non-diagonal weight-matrix. We show that in all the examined cases the uncertainties in the velocities computed taking into account the detailed noise spectrum are larger by a factor of 4 ± 1 than the formal uncertainties obtained by the least-squares method under the assumption of pure white noise. Estimating the slope of a time-series taking into account the autocorrelation of the samples yields velocities not significantly different from those obtained assuming uncorrelated samples. We conclude that the reason for the velocity uncertainty estimated by the standard least-squares method being unrealistically small is the neglect of the cumulative effect of uncorrelated and correlated noise. Neglecting the correlated noise does not, however, affect the velocity. Earlier investigations based on more limited (<3 yr) data sets have resulted in non-unique conclusions as to the time decrease of the velocity uncertainty with time, and the noise spectrum of the time-series. We find that the velocity uncertainty does decrease as the time-series increases, and the value of the velocity uncertainty can be predicted from the power spectral density, as the length of the time-series increases. The time-series are finally analysed in the space domain. After removal of common errors, typically represented by sinusoids of annual period, correlation coefficients are computed for pairs of stations and plotted as a function of their distance. We find that the time-series already decorrelate at very short distances (<100 km). This suggests that random errors affecting the coordinates of clusters of stations such as, for example, atmospheric refraction or mismodelling of the orbits are negligible in our time-series. The estimates of the velocities and uncertainties of the permanent stations obtained by spectral analysis form the basis for a subsequent investigation of the present-day, large-scale strain rate field in the Alpine Mediterranean area, which is implied by these scattered surface displacements.