The effect of coloured noise on the uncertainties of rates estimated from geodetic time series

Until recently, it was typically assumed that only white noise was present in geodetic time series. However, several data sets have now provided evidence for the presence of power-law noise. The uncertainty of any rates estimated from such data sets is dependent on the error model assumed for the data. Here a general form for the covariance matrix for any power-law noise model is introduced and simple equations that relate the rate uncertainty to the noise amplitude, sampling frequency and length of the time series are empirically derived. In addition, equations to analyse data sets and obtain a rate uncertainty when computational speed is at a premium are provided. These equations are tested against previously published geodetic data sets.