To estimate the velocities of the ISGPS stations a standard weighted least squares approach is used to estimate the best line fit to each coordinate component of the data along with the associated errors. We model the observations of the coordinate component y as a linear function of time, yi=ati + b + ei, where yi are the measurements (displacements) made at times ti and ei are the errors, assumed to be normally distributed. The parameter a is the average velocity of the station in each coordinate component. To simplify the following equations we let and introduce the matrix
We estimate the covariance matrix of the parameters a and b as
If we write out the equation for the velocity uncertainty, the square root of the
upper left diagonal in P (equation 8), we obtain
The velocity uncertainty (equation 10) is obtained assuming that the noise in the GPS data is normally distributed and uncorrelated in time. However, the noise characteristics of GPS time series are correlated in time [Langbein et al. (1997)]. [Mao et al. (1999)] find a combination of white noise and flicker noise to be the best model for the noise characteristics. They state that the velocity uncertainty derived from GPS coordinate time series may be underestimated by factors of 5-11 if a pure white noise model is assumed. In this study the velocity uncertainties are estimated as described in equation 10. We expect that the uncertainties may be too small. A more rigorous uncertainty estimate will be carried out in later studies.
Another estimator for the quality of fit is the parameter . The weighted residual sum of squares is
The values of are strongly dependent on which scaling factors are used for the formal coordinate errors. From equations 6 and 12 it is obvious that if the coordinate errors are scaled by a factor s, then scales as 1/s2.