## 1. Introduction

[2] Continuous Global Positioning System (GPS) measurements have been used now nearly 15 years for estimation of crustal deformation. Station positions are determined with respect to an earth-fixed terrestrial reference system. Geophysical studies using geodetic measurements of surface displacement or strain require not only accurate estimates of these parameters but also accurate error estimates. The precision of these estimates is often assessed by their repeatability defined by the mean squared error of individual coordinate components (i.e., north, east, and vertical) about a linear trend. Except for the significant episodic deformation, such as large earthquakes, a linear trend can be a good representative of the deformation behavior. The site velocities are usually determined by linear regression of individual coordinate components. The least squares technique is used to estimate the line parameters, i.e., the intercept and the slope (site velocity).

[3] In the ideal case, it is desired that the time series possess only white noise and all functional effects are fully understood. The noise in GPS coordinate time series turns out not to be white. Several geodetic data sets have provided evidence for error sources that introduce large temporal correlations into the data. The ultimate goal of noise studies is to come up with a stochastic model that allows one to process the coordinate time series such that the “best” solution (most precise solution together with proper precision description) of the station positions and site velocities can be determined. An intermediate goal is therefore to better understand and to identify the various noise components of the stochastic model.

[4] Two techniques have generally been employed to assess the noise characteristics of geodetic time series, namely, the power spectral method and the maximum likelihood estimation (MLE) method. The former is aimed to examine the data in the frequency domain while the latter is used to examine the data covariance matrix in the time (space) domain. The MLE can estimate the parameters of a noise model effectively in contrast to the classical power spectra techniques. In this contribution, we will not make use of the spectral techniques. The MLE method is generally used to compute the amount of white noise, flicker noise, and random walk noise in the time series [see, e.g., *Zhang et al.*, 1997; *Langbein and Johnson*, 1997; *Mao et al.*, 1999; *Williams et al.*, 2004; *Langbein*, 2004]. In this paper, we introduce and use a different variance component estimation method based on the least squares principle. Our motivation is given in the next section.