## 1. Introduction

[2] Several papers, including *Langbein and Johnson* [1997], *Zhang et al.* [1997], *Mao et al.* [1999], *Williams et al.* [2004], and *Beavan* [2005], have demonstrated that daily, geodetic measurements of position or distance changes are temporally correlated rather than simply independent observations. The first-order effects of temporally correlated data have been summarized by *Johnson and Agnew* [1995], *Williams* [2003], and *Langbein* [2004]. They show that using models describing temporal correlations directly affects the estimates of the standard error of the rates that are derived by using least squares fitting a linear trend in time to the deformation time series. The model that describes the temporal correlations of the data is quantified in the data covariance matrix. The covariance matrix is used in conjunction with a function to fit, in a least squares sense, the temporal variations, including the rate, to the time series of the deformation data [e.g., *Menke*, 1984]. The covariance matrix represents the assumed noise processes of the data.

[3] The noise model for geodetic data has been modeled in the frequency domain as a combination of white noise, where the power density is independent of frequency, and a power law, *f*^{−n}, where *f* is frequency. Initial characterization of the power law process was done by *Langbein and Johnson* [1997] using two-color electronic distance meter (EDM) observations, and restricted the power law index, *n*, to be 2. They assumed that the temporally correlated process for noise was that of a random walk, which might characterize localized, random motions of the geodetic monument [*Wyatt*, 1982, 1989]. However, studies by *Zhang et al.* [1997] and *Mao et al.* [1999] of the time series of position changes measured by GPS suggested that the appropriate process for GPS was that of flicker noise, where *n* = 1.

[4] *Williams et al.* [2004] did a comprehensive analysis of over 400 GPS stations, and for those sites that are from regional networks and from which common-mode signals had been removed, the noise models for the data were best characterized by power law noise, where the index was between that of flicker and random walk. In many cases, since the *n* ≈ 1, there was no compelling reason to reject the hypothesis of a flicker-noise process. In addition, *Williams et al.* [2004] explored the relationship between monument design and the level of noise. They concluded that the deeply braced monuments [*Wyatt et al.*, 1989; *Langbein et al.*, 1995] employed by the Southern California Integrated GPS Network (SCIGN) had less temporally correlated noise than other types of monuments used in that network. In addition, the level of correlated noise from the Basin and Range Geodetic Network (BARGEN) was significantly less than the other three regional networks that they analyzed. The BARGEN network can be considered homogeneous in terms of monument type, all being deeply braced monuments; and environment, with all sites located in the desert with low erosion rates and low humidity, both factors that might lessen the effect of correlated noise.

[5] *Beavan* [2005] re-examined the monument stability problem by comparing the noise models derived from time series of position changes estimated from GPS measurements from massive, cement pillars used in New Zealand with results obtained from the three US regional networks analyzed by *Williams et al.* [2004]. He concluded that the correlated noise in the New Zealand data did not differ significantly from those of the US networks, which included a mix of monument types, including many deeply braced monuments.

[6] *Langbein* [2004] extended the possible set of noise models and re-examined the two-color EDM data discussed by *Langbein and Johnson* [1997]. In addition to the power law noise, he derived covariance matrices that incorporated Gauss-Markov processes, and band-passed filtered noise which could be used to characterize the seasonal component of noise found in many geodetic time series. In addition to the noise modeling, for the deterministic part of the analysis, which included rate, rate changes, offsets due to earthquakes or site maintenance, *Langbein et al.* [2006] extended the list of functions to characterize postseismic deformation, including Omori's Law, where displacement is proportional to log (1 + *t*/*τ*) and an exponential decay, where displacement is proportional to 1−*e*^{−t/τ}.

[7] For the two-color EDM data, *Langbein* [2004] found that no single noise model satisfied all of the data. Rather, he found that all of the models described above were needed to characterize the noise in the EDM observations. In contrast to the noise models derived for GPS data by *Williams et al.* [2004], the noise in the EDM time series appears to be more complex. In part, this difference is due to the fact that the more complex noise models have not been applied to the GPS observations and, secondly, the EDM data provided, on average, a longer time series for which the longer period components of the noise could be resolved [*Williams et al.*, 2004; *Langbein*, 2004].

[8] This report tests whether the more complex noise models derived by *Langbein* [2004] provide a significantly better fit to the GPS observations than the power law model. In addition, since several years have elapsed since the *Williams et al.* [2004] study, the GPS time series used here are longer than those available to *Williams et al.* [2004]. Where *Williams et al.* [2004] used time series from the SCIGN array that, on average had 2.5 years of observation, this report restricts the length to greater than 3.9 years; The longer time series, which were not available to *Williams et al.* [2004], should help resolve the longer period components of the noise.