## 1. Introduction

[2] Velocities derived from geodetic coordinate time series are now routinely used as input to geophysical models [*Segall and Davis*, 1997], with many applications including plate boundary dynamics, postglacial rebound, surface mass loading, and global sea level change. It has recently emerged that GPS coordinate time series have significant annual signals [e.g., *Van Dam et al.*, 2001], which might significantly bias published velocity estimates. This paper systematically investigates the effect of annual signals on geodetic velocities and complements recent research on power law noise of coordinate time series [*Zhang et al.*, 1997; *Mao et al.*, 1999]. Indeed, seasonally driven signals with an annually repeating component would include power at all annual harmonic frequencies, which would affect both spectral and time domain characterizations of GPS errors.

[3] A major component of annual signals is now known to be true physical site motion. The dominant cause for annual signals with respect to a global reference frame is surface loading due to hydrology and atmospheric pressure. *Van Dam et al.* [2001, p. 651] report that in loading models of continental water storage, “vertical displacements have a root mean square (RMS) values as large as 8 mm and are predominantly annual in character.” These hydrological models (also accounting for atmospheric loading) were shown to correlate strongly with the same GPS coordinate time series used here, with the variance reduction in GPS height residuals being approximately equal to the variance of the model. Hence seasonal variation, which is best described by a deterministic model (rather than a power law noise model), is likely to contribute to velocity error for globally referenced coordinates, especially over short data spans. Until physical models of annual signals adequately describe the observed variation, a reasonable solution to this problem is to estimate an annual signal (amplitude and phase) simultaneously with site velocities and initial positions. Another strategy is to reference coordinates regionally (e.g., by spatial filtering [*Wdowinski et al.*, 1997]), which would not be effective for large regions such as the North American–Pacific plate boundary, or for stability tests of major plates. A major component of annual signals is now known to be true physical site motion [*Blewitt et al.*, 2001].

[4] There are several concerns motivating our research. First, there are numerous recent examples of published estimates of tectonic velocities with as little as 1–2 year data spans, where annual signals have not been taken into account in the estimated velocities and errors. Their effects have often been ignored completely or have been subject to incorrect intuitive speculation. For example, *Dixon and Mao* [1997, p. 536] state “the influence of annual errors … on velocity estimates would be minimal for an integer number of years but would affect velocity estimates for the 2.5 year time span used here” (which our research here proves to be wrong). If annual signals are not taken into account, it is shown here that they typically dominate velocity errors during the first 2.5 years of coordinate time series, with the nonobvious theoretical result supported by our data that the bias drops rapidly between 2 and 2.5 years. This is consistent with anecdotal evidence that GPS velocity solutions tend to be unstable until a 2-year data span is exceeded. Our research here can be used as a guide to set criteria for publishing new results and can also be used to assess the level of errors in previously published results.

[5] Second, geodetic investigations almost always rely on the availability of an accurate global reference frame, either directly or indirectly. For example, such a frame is essential for accurate orbits produced by the International GPS Service (IGS), Earth rotation parameters produced by the International Earth Rotation Service (IERS), globally referenced site coordinates, and global site velocities to define kinematics relative to stable plate interiors for deformation studies. Current procedures to produce the IERS Terrestrial Reference Frame (ITRF) do not account for annual signals in deriving site velocities (though future versions may do so if this type of research demonstrates the benefits). To consider the effects on current ITRF and IGS procedures, it should be kept in mind that such analyses assume no time correlation between epoch solutions; in fact this class of solution currently dominates the tectonophysics literature. It should also be noted that even if power law stochastic models (e.g., flicker noise) were used, they alone would not account properly for time domain behavior due to annual signals.

[6] Third, annually repeating signals generally contain not only an annual sinusoidal component but also the annual harmonics. Estimation of only the annual amplitude and phase will therefore not mitigate the entire effect of an annually repeating signal. It is not immediately obvious how many extra terms should be included.

[7] Fourth, while estimation of annual harmonics may reduce systematic error, it will also introduce a greater random error in velocity due to the increased number of parameters. We can expect this to be a problem for shorter time series, when correlations between the estimated parameters become increasingly significant. We might expect that below some minimum data span the systematic bias we are attempting to mitigate would be less harmful than the dilution of velocity precision.

[8] Guided by these concerns, we formulate the following research questions. First, there are fundamental questions: What are the temporal characteristics of velocity bias in the presence of an annually repeating signal when the velocity estimation assumes no deterministic model or interepoch correlations? Can such temporal characteristics be used to advantage? Second, there are questions of interpretation of published results: How should we interpret errors of published velocity solutions (and hence the significance of the research findings) that have not accounted for annual signals? What is the minimum data span at which one should accept velocity estimates (e.g., as input to ITRF) that have not accounted for annual signals? Third, there are questions of implementation. What is the data span beyond which the degradation in precision arising from estimation of extra annual signal parameters is smaller than typical systematic bias? What is the data span beyond which negligible gain is to be made by estimating annual (and harmonic) signals?

[9] We begin by systematically developing a theoretical foundation for the analysis of annually repeating signals and their effect on velocity. We proceed to develop an error model that might be used to reinterpret errors of published results; this is tested using our own data. On the basis of the theory and data we then answer the research questions posed above.