## 1. Introduction

[2] The study of noise in geodetic time series has been pursued for a number of important reasons. Understanding noise in the time series is vital for the detection and interpretation of the signals of interest. It is especially important in understanding the uncertainties of parameters estimated from the time series [e.g., *Langbein and Johnson*, 1997; *Zhang et al.*, 1997; *Beavan*, 2005; *Langbein*, 2008; *Santamaría-Gómez et al.*, 2011]. Noise can also reveal shortcomings in the models and techniques used for the underlying analysis from which the time series are obtained [e.g., *Penna and Stewart*, 2003; *Penna et al.*, 2007; *King et al.*, 2008; *King and Watson*, 2010].

[3] The “noise” in the time series is defined to be the residual signal relative to a model that is estimated prior to or simultaneous with the noise analysis. This model generally includes a seasonal signal, for seasonal signals are present in a variety of geodetic time series. In some cases, the seasonal signals reflect geophysical signal. For example, some Global Navigation Satellite Systems (GNSS) sites experience a real seasonal motion associated with local environmental effects, such as rain [e.g., *King et al.*, 2007], temperature [e.g., *Prawirodirdjo et al.*, 2006], surface loading [e.g., *van Dam et al.*, 2001], and aquifer pumping [e.g., *Bell et al.*, 2002]. Other environmental effects do not involve actual motion of the antenna phase center. For example, multipath/scattering [e.g., *Elósegui et al.*, 1995] is not modeled in the phase solutions and thereby induces a systematic error in the position estimate that appears as correlated noise in the time series [e.g., *Park et al.*, 2004; *King and Watson*, 2010].

[4] The seasonal signal is typically represented by sums of sinusoids with annual frequency and its harmonics. Whether this seasonal signal represents systematic error (the elimination of which by the development of improved models is always preferable) or true climatic signal, it is important to model this seasonal signal since it can impact parameters of interest estimated from the time series, particularly the site velocity [e.g., *Blewitt and Lavallée*, 2002; *Bos et al.*, 2010].However, there is reason to expect that the seasonal signal in geodetic time series (itself being a response to environmental changes) is not time-invariant. Environmental variables are known to have a red power spectrum and are typically expressed in terms of an inverse power law [e.g.,*Vasseur and Yodzis*, 2004]. Biological populations, for example, are known to exhibit a red spectrum in response to environmental noise that underlies seasonal variability [e.g., *Halley and Inchausti*, 2004].

[5] Several previous analyses have taken interseasonal variability into account. *Murray and Segall* [2005]modeled the seasonal amplitudes as a random walk process and used a Kalman filter to estimate the time-dependent parameters, an approach used in the study of*Wernicke and Davis* [2010] as well as here. *Davis et al.* [2006] used piecewise continuous linear polynomials to represent the seasonal amplitudes, and *Bennett* [2008] employed a more general representer method.

[6] Recent studies have identified periodic noise in GPS time series at harmonics of the GPS draconitic frequency [e.g., *Barrett*, 2008; *Ray et al.*, 2008; *Tregoning and Watson*, 2009; *King and Watson*, 2010]. The draconitic period for the GPS constellation is ∼351.4 days [*Tregoning and Watson*, 2009], making the draconitic frequency ∼1.04 cycles per year (cpy). Thus, an individual time series would have to have a length of ∼25 years to separate noise at the fundamental draconitic frequency and the climatic seasonal frequency of 1 cpy. Noise in the draconitic spectrum is therefore identified by stacking power spectra of GPS time series. Near the lower harmonics (annual and semiannual), climatic seasonal signals appear to dominate the error spectrum [*Barrett*, 2008; *Ray et al.*, 2008], although both unmodeled multipath [*King and Watson*, 2010] and atmospheric loading contributions at tidal periods [*Tregoning and Watson*, 2009] have been shown to yield noise at the draconitic annual and semiannual frequency. In this study we assume that the term “seasonal” applies to climatic seasonal signals, and we leave to future study the problem of separation of climatic and draconitic signals.

[7] Below, we explore the implications for modeling and noise analysis of stochastic seasonal processes of climatic origin in geodetic time series. We investigate the form of the power spectral density of a time series having a stochastic seasonal component. We then model this seasonal variability using a Kalman filter in two types of geodetic time series. The first is a time series of the vertical coordinate of site position of a GNSS site. We also consider a time series of surface mass calculated from Gravity Recovery and Climate Experiment (GRACE) data, wherein the observed seasonal signal is greater than the variability in the rate. We begin by developing a theory for expressing this variability, focusing on the annual signal.