## 1 Introduction

[2] Horizontal and vertical velocities of permanent GPS stations are commonly estimated from the available position time series. Proper analysis of position time series is of particular interest for many geophysical applications that require unbiased estimates of velocities and their uncertainties. There are several issues known to affect the estimation of site velocity and its uncertainty. The ultimate goal of the GPS position time series studies is to discriminate between the functional and the stochastic effects in the series. Both effects are relevant in geophysical phenomena and hence the subject of the present contribution. Functional effects, such as a linear trend, offsets, and potential periodicities, can be well explained by a deterministic model, while the remaining unmodeled effects can be described by a proper stochastic model. Both models should optimally be selected and analyzed for proper analysis of the time series.

[3] Seasonal variations in site positions consist of signals from various geophysical sources and systematic modeling errors [*Dong et al*., 2002]. They showed that 40% of the seasonal power can be explained by redistributions of geophysical fluid mass loads. *Penna and Stewart* [2003] and *Stewart et al*. [2005] showed how unmodeled periodic errors at tidal frequencies can result in spurious longer periodic effects in the resultant time series. *Yuan et al*. [2009] reported that the K2 body tide and ocean tide loading periodic errors can significantly bias the site velocities. *Amiri-Simkooei et al*. [2007] showed that the seasonal variations, to a large extent, can be modeled by a set of harmonic functions. The colored noise of the series was shown to mimic periodic patterns, and therefore it could be compensated by using a set of harmonic functions. The results from *Ray et al*. [2008], *Collilieux et al*. [2007], *Amiri-Simkooei et al*. [2007], *Tregoning and Watson* [2009], *King and Watson* [2010], and *Santamaría-Gómez et al*. [2011] revealed harmonics of around 351 days and its higher harmonics in the series, which coincides with the “GPS draconitic year” period, i.e., the 351.4 days required for a GPS orbit to repeat its inertial orientation with respect to the Sun. This periodic pattern has been observed in the power spectra of nearly all products of the IGS [*Griffiths and Ray*, 2013]. The possible causes for these draconitic signals lie in spurious aliasing, orbital errors (e.g., solar radiation modeling or eclipse modeling) and/or propagation of site-dependent effects such as multipath.

[4] The temporal noise characteristics of GPS time series is well described as a combination of white noise and power law noise [*Zhang et al*., 1997; *Williams et al*., 2004]. Spatial correlation between time series is also considered to be significant [*Williams et al*., 2004]. *Amiri-Simkooei* [2009] presents a multivariate noise assessment of GPS time series in which both time and space correlated noise components are simultaneously estimated using the least squares variance component estimation [*Teunissen and Amiri-Simkooei*, 2008; *Amiri-Simkooei*, 2007]. In the studies of *Ray et al*. [2008] and *Amiri-Simkooei et al*. [2007], the power spectrum of the series has been estimated on the assumption that temporal and spatial correlation are absent. We now consider these issues on temporal and spatial noise components to optimally estimate the power spectrum density using a multivariate analysis. The presence of colored noise as well as the small amplitude of the signals makes the detection power of the signals inefficient. This is because color noise can also mimic periodic variations [*Amiri-Simkooei et al*., 2007; *Williams*, 2007], and hence it can be mixed with real periodic patterns. Multivariate analysis solves these issues.

[5] This paper is organized as follows. In section 2, we derive the formulation of the harmonic estimation for a multivariate linear model. The goal is to detect common-mode signals that are assumed to be present in all GPS position time series. To detect significant signals in GPS time series, the existing spectral analysis methods (e.g., power spectra estimated using the Lomb-Scargle periodogram) are usually formulated based on the assumption that the time series have only white noise and that they are uncorrelated to each other. We give an extension of the least squares harmonic estimation for a multivariate and multiharmonic model, which includes both temporal and spatial correlation of the series. Section 3 applies this theory to daily position time series of 350, 150, and 50 permanent GPS stations. The multivariate power spectrum of the series is estimated using the multivariate harmonic estimation model. Such a model will also provide us with the time- and space-correlated noise of GPS time series. We provide some useful observations on the nature of the periodic pattern reported by *Amiri-Simkooei* [2007], *Ray et al*. [2008], *Collilieux et al*. [2007], *King and Watson* [2010], and *Santamaría-Gómez et al*. [2011]. A (more) precise estimate of the period of this signal along with its spatial variations is highlighted in the multivariate analysis. Further, in plate tectonics studies, for an unbiased site velocity estimation and a realistic assessment of its uncertainty, it is required to compensate for such a significant periodic effect in the functional part of the model using a series of sinusoidal functions.