Velocity estimation

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, y_i=at_i + b + e_i, where y_i are the measurements (displacements) made at times t_i and e_i 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 ${\bf\beta} = (b \quad a)^{T}$ and introduce the matrix

$$M= \left( \begin{array}{cc} 1 &t_1 \\ 1 &t_2 \\ \vdots & \vdots \\ 1 &t_N \end{array} \right) ,$$ (5)

where N is the number of available measurements. Thus we can express the data as ${\bf y} = M{\bf\beta} + {\bf e}$. To weigh the observations correctly we define the weight matrix W as the diagonal of the inverse square of the rescaled coordinate uncertainties $\sigma_{i}^{crd}$ (see Section 3.1):

$$W=\left( \begin{array}{cccc} \frac{1}{(\sigma_{1}^{crd})^{2}... ...ots &0 &\frac{1}{(\sigma_{N}^{crd})^{2}} \end{array} \right).$$ (6)

According to least squares theory the best estimate of $\bf {\beta}$ is then [Press et al. (1992)]:

$${\bf\beta}= ((M^{T}WM)^{-1}M^{T}W){\bf y}.$$ (7)

We estimate the covariance matrix of the parameters a and b as [Brockmann (1997)]:

$$P = \hat{\sigma}_{crd}^{2} \left( \begin{array}{cc} \disp... ...splaystyle \sum_{i=1}^{N} t_{i} & N \end{array} \right)^{-1}$$ (8)

where $\hat{\sigma}_{crd}^{2}$ is the variance of the residuals:

$$\hat{\sigma}_{crd}^{2} = \frac{ \displaystyle \sum_{i=1}^{N} (y_i - (at_i + b))^2 } {N-2}.$$ (9)

If we write out the equation for the velocity uncertainty, the square root of the upper left diagonal in P (equation 8), we obtain

$$\sigma^{vel} = \hat{\sigma}_{crd} \sqrt{ \frac{1}{\displays... ...} - \left( \displaystyle \sum_{i=1}^{N} t_{i} \right)^2 }. }$$ (10)

This tells us that the velocity uncertainty estimate is proportional to the standard deviation of the residual and independent of the coordinate errors $\sigma_{i}^{crd}$. We can always shift the time scale so that $\sum_{i=1}^{N} t_{i}$ becomes zero, and equation 10 then becomes

$$\sigma^{vel} = \hat{\sigma}_{crd} \sqrt{\frac{1}{ \displaystyle \sum_{i=1}^{N} t_{i}^{2} } }.$$ (11)

The sum $\sum_{i=1}^{N} t_{i}^{2}$ increases as N³ so the velocity uncertainty decreases roughly as N^-1.5.

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 $\chi_{\nu}^{2}$. The weighted residual sum of squares is

$$WRSS=\bf {r}^{T} W \bf {r}.$$ (12)

The number of degrees of freedom for linear regression is N - q, where N is the number of data used in each best line fit and q is the number of unknown parameters (q=2). The parameter $\chi_{\nu}^{2}$ is then defined as

$$\chi_{\nu}^{2} = WRSS/(N-2).$$ (13)

A $\chi_{\nu}^{2}=1$ indicates that the model fits the data perfectly and that the sizes of the coordinate errors are appropriate. If $\chi_{\nu}^{2} > 1$ then either the model does not represent the data very well, or the coordinate errors $\sigma^{crd}_{i}$ are too small, assuming normally distributed errors. A $\chi_{\nu}^{2} < 1$ indicates that the coordinate errors may be overestimated.

The values of $\chi^{2}_{\nu}$ 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 $\chi^{2}_{\nu}$ scales as 1/s².